Theorem cdeqal1

Description: Distribute conditional equality over quantification. Usage of this theorem is discouraged because it depends on ax-13 . (Contributed by Mario Carneiro, 11-Aug-2016) (New usage is discouraged.)

	Ref	Expression
Hypothesis	cdeqnot.1	|- CondEq ( x = y -> ( ph <-> ps ) )
Assertion	cdeqal1	|- CondEq ( x = y -> ( A. x ph <-> A. y ps ) )

Proof

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