Theorem cdeqal1 2820

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 2815	. . 3 |- ( x = y -> ( ph <-> ps ) )
3	2	cbvalv 1839	. 2 |- ( A. x ph <-> A. y ps )
4	3	cdeqth 2816	1 |- CondEq ( x = y -> ( A. x ph <-> A. y ps ) )

Syntax hints:   <-> wb 103   A.wal 1285  CondEqwcdeq 2812

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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470

This theorem depends on definitions:  df-bi 115  df-nf 1393  df-cdeq 2813

This theorem is referenced by: (None)