Theorem cdeqal 3040

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 3037	. . 3 |- ( x = y -> ( ph <-> ps ) )
3	2	albidv 1877	. 2 |- ( x = y -> ( A. z ph <-> A. z ps ) )
4	3	cdeqi 3036	1 |- CondEq ( x = y -> ( A. z ph <-> A. z ps ) )

This proof depends on syntax axioms:   <-> wb 105   A.wal 1400  CondEqwcdeq 3034

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-17 1579

This proof depends on definitions:  df-bi 117  df-cdeq 3035

This theorem is used by: (None)