Theorem ceqsal

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993)

	Ref	Expression
Hypotheses	ceqsal.1	|- F/ x ps
	ceqsal.2	|- A e. _V
	ceqsal.3	|- ( x = A -> ( ph <-> ps ) )
Assertion	ceqsal	|- ( A. x ( x = A -> ph ) <-> ps )

Proof

Step	Hyp	Ref	Expression
1		ceqsal.1	|- F/ x ps
2		ceqsal.2	|- A e. _V
3		ceqsal.3	|- ( x = A -> ( ph <-> ps ) )
4	1 3	ceqsalg	|- ( A e. _V -> ( A. x ( x = A -> ph ) <-> ps ) )
5	2 4	ax-mp	|- ( A. x ( x = A -> ph ) <-> ps )