Theorem cdeqal 2804

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

Hypothesis

Ref	Expression
cdeqnot.1	|- CondEq ( x = y -> ( ph <-> ps ) )

Assertion

Ref	Expression
cdeqal	|- CondEq ( x = y -> ( A. z ph <-> A. z ps ) )

Distinct variable groups:   x, z   y, z

Allowed substitution hints:   ph( x, y, z)   ps( x, y, z)

Proof of Theorem cdeqal

Step	Hyp	Ref	Expression
1		cdeqnot.1	. . . 4 |- CondEq ( x = y -> ( ph <-> ps ) )
2	1	cdeqri 2801	. . 3 |- ( x = y -> ( ph <-> ps ) )
3	2	albidv 1745	. 2 |- ( x = y -> ( A. z ph <-> A. z ps ) )
4	3	cdeqi 2800	1 |- CondEq ( x = y -> ( A. z ph <-> A. z ps ) )

Syntax hints:   <-> wb 103   A.wal 1282  CondEqwcdeq 2798

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1376  ax-gen 1378  ax-17 1459

This theorem depends on definitions:  df-bi 115  df-cdeq 2799

This theorem is referenced by: (None)