Theorem cdeqab

Description: Distribute conditional equality over abstraction. (Contributed by Mario Carneiro, 11-Aug-2016)

	Ref	Expression
Hypothesis	cdeqnot.1	$$⊢CondEq\left(x=y\to \left(\phi \leftrightarrow \psi \right)\right)$$
Assertion	cdeqab	$$⊢CondEq\left(x=y\to \left\{z|\phi \right\}=\left\{z|\psi \right\}\right)$$

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	abbidv	$$⊢x=y\to \left\{z|\phi \right\}=\left\{z|\psi \right\}$$
4	3	cdeqi	$$⊢CondEq\left(x=y\to \left\{z|\phi \right\}=\left\{z|\psi \right\}\right)$$