Theorem cdeqal1 3426

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 3421	. . 3 |- ( x = y -> ( ph <-> ps ) )
3	2	cbvalv 2273	. 2 |- ( A. x ph <-> A. y ps )
4	3	cdeqth 3422	1 |- CondEq ( x = y -> ( A. x ph <-> A. y ps ) )

Syntax hints:   <-> wb 196   A.wal 1481  CondEqwcdeq 3418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cdeq 3419

This theorem is referenced by: (None)