Editing Universal quantification (section)

=== Other connectives ===
The universal (and existential) quantifier moves unchanged across the [[logical connective]]s [[logical conjunction|∧]], [[logical disjunction|∨]], [[material conditional|→]], and [[converse nonimplication|↚]], as long as the other operand is not affected;<ref>that is, if the variable <math>y</math> does not occur free in the formula <math>P(x)</math> in the equivalences below</ref> that is:

:<math>\begin{align}
P(x) \land (\exists{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \land Q(y)) \\
P(x) \lor  (\exists{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \lor Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \to   (\exists{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \to Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \nleftarrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \nleftarrow Q(y)) \\
P(x) \land (\forall{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \land Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \lor  (\forall{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \lor Q(y)) \\
P(x) \to   (\forall{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \to Q(y)) \\
P(x) \nleftarrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) &\equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \nleftarrow Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset
\end{align}</math>

Conversely, for the logical connectives [[Sheffer stroke|↑]], [[Logical NOR|↓]], [[Material nonimplication|↛]], and [[converse implication|←]], the quantifiers flip:

:<math>\begin{align}
P(x) \uparrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \uparrow Q(y)) \\
P(x) \downarrow  (\exists{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \downarrow Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \nrightarrow (\exists{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \nrightarrow Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \gets (\exists{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \forall{y}{\in}\mathbf{Y}\, (P(x) \gets Q(y)) \\
P(x) \uparrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \uparrow Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
P(x) \downarrow  (\forall{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \downarrow Q(y)) \\
P(x) \nrightarrow (\forall{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \nrightarrow Q(y)) \\
P(x) \gets (\forall{y}{\in}\mathbf{Y}\, Q(y)) & \equiv\ \exists{y}{\in}\mathbf{Y}\, (P(x) \gets Q(y)),& \text{provided that } \mathbf{Y}\neq \emptyset \\
\end{align}</math>
<!-- What about:
*[[logical biconditional|Biconditional (if and only if) (xnor)]] (<math>\leftrightarrow</math>, <math>\equiv</math>, or <math>=</math>)
*[[Exclusive or|Exclusive disjunction (xor)]] (<math>\not\leftrightarrow</math>)
-->