Theorem ceqsalgALT

Description: Alternate proof of ceqsalg , not using ceqsalt . (Contributed by NM, 29-Oct-2003) (Proof shortened by Andrew Salmon, 8-Jun-2011) (Revised by BJ, 29-Sep-2019) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Hypotheses	ceqsalg.1	|- F/ x ps
	ceqsalg.2	|- ( x = A -> ( ph <-> ps ) )
Assertion	ceqsalgALT	|- ( 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		elisset	|- ( A e. V -> E. x x = A )
4		nfa1	|- F/ x A. x ( x = A -> ph )
5	2	biimpd	|- ( x = A -> ( ph -> ps ) )
6	5	a2i	|- ( ( x = A -> ph ) -> ( x = A -> ps ) )
7	6	sps	|- ( A. x ( x = A -> ph ) -> ( x = A -> ps ) )
8	4 1 7	exlimd	|- ( A. x ( x = A -> ph ) -> ( E. x x = A -> ps ) )
9	3 8	syl5com	|- ( A e. V -> ( A. x ( x = A -> ph ) -> ps ) )
10	2	biimprcd	|- ( ps -> ( x = A -> ph ) )
11	1 10	alrimi	|- ( ps -> A. x ( x = A -> ph ) )
12	9 11	impbid1	|- ( A e. V -> ( A. x ( x = A -> ph ) <-> ps ) )