Theorem cdeqim 2955

Description: Distribute conditional equality over implication. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypotheses

Ref	Expression
cdeqnot.1	|- CondEq ( x = y -> ( ph <-> ps ) )
cdeqim.1	|- CondEq ( x = y -> ( ch <-> th ) )

Assertion

Ref	Expression
cdeqim	|- CondEq ( x = y -> ( ( ph -> ch ) <-> ( ps -> th ) ) )

Proof of Theorem cdeqim

Step	Hyp	Ref	Expression
1		cdeqnot.1	. . . 4 |- CondEq ( x = y -> ( ph <-> ps ) )
2	1	cdeqri 2948	. . 3 |- ( x = y -> ( ph <-> ps ) )
3		cdeqim.1	. . . 4 |- CondEq ( x = y -> ( ch <-> th ) )
4	3	cdeqri 2948	. . 3 |- ( x = y -> ( ch <-> th ) )
5	2, 4	imbi12d 234	. 2 |- ( x = y -> ( ( ph -> ch ) <-> ( ps -> th ) ) )
6	5	cdeqi 2947	1 |- CondEq ( x = y -> ( ( ph -> ch ) <-> ( ps -> th ) ) )

Syntax hints:   -> wi 4   <-> wb 105  CondEqwcdeq 2945

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117  df-cdeq 2946

This theorem is referenced by: (None)