Theorem cdeqal1

Description: Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 11-Aug-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cdeqnot.1	$$⊢CondEq\left(x=y\to \left(\phi \leftrightarrow \psi \right)\right)$$
Assertion	cdeqal1	$$⊢CondEq\left(x=y\to \left(\forall x\phi \leftrightarrow \forall y\psi \right)\right)$$

Proof

Step	Hyp	Ref	Expression
1		cdeqnot.1	$$⊢CondEq\left(x=y\to \left(\phi \leftrightarrow \psi \right)\right)$$
2	1	cdeqri	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
3	2	cbvalv	$$⊢\forall x\phi \leftrightarrow \forall y\psi$$
4	3	cdeqth	$$⊢CondEq\left(x=y\to \left(\forall x\phi \leftrightarrow \forall y\psi \right)\right)$$