Theorem cdeqal1 2946

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
cdeqal1	|- CondEq ( x = y -> ( A. x ph <-> A. y ps ) )

Distinct variable groups:   ps, x   ph, y

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

Proof of Theorem cdeqal1

Step	Hyp	Ref	Expression
1		cdeqnot.1	. . . 4 |- CondEq ( x = y -> ( ph <-> ps ) )
2	1	cdeqri 2941	. . 3 |- ( x = y -> ( ph <-> ps ) )
3	2	cbvalv 1910	. 2 |- ( A. x ph <-> A. y ps )
4	3	cdeqth 2942	1 |- CondEq ( x = y -> ( A. x ph <-> A. y ps ) )

Syntax hints:   <-> wb 104   A.wal 1346  CondEqwcdeq 2938

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-cdeq 2939

This theorem is referenced by: (None)