Theorem ceqsalg

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. For an alternate proof, see ceqsalgALT . (Contributed by NM, 29-Oct-2003) (Proof shortened by BJ, 29-Sep-2019)

	Ref	Expression
Hypotheses	ceqsalg.1	|- F/ x ps
	ceqsalg.2	|- ( x = A -> ( ph <-> ps ) )
Assertion	ceqsalg	|- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )

Proof

Step	Hyp	Ref	Expression
1		ceqsalg.1	|- F/ x ps
2		ceqsalg.2	|- ( x = A -> ( ph <-> ps ) )
3	2	ax-gen	|- A. x ( x = A -> ( ph <-> ps ) )
4		ceqsalt	|- ( ( F/ x ps /\ A. x ( x = A -> ( ph <-> ps ) ) /\ A e. V ) -> ( A. x ( x = A -> ph ) <-> ps ) )
5	1 3 4	mp3an12	|- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )