Theorem cdeqal1 3019

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 3014	. . 3 |- ( x = y -> ( ph <-> ps ) )
3	2	cbvalv 1964	. 2 |- ( A. x ph <-> A. y ps )
4	3	cdeqth 3015	1 |- CondEq ( x = y -> ( A. x ph <-> A. y ps ) )

Syntax hints:   <-> wb 105   A.wal 1393  CondEqwcdeq 3011

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580

This theorem depends on definitions:  df-bi 117  df-nf 1507  df-cdeq 3012

This theorem is referenced by: (None)