Theorem ceqsexg 3181

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 11-Oct-2004.)

Hypotheses

Ref	Expression
ceqsexg.1	|- F/ x ps
ceqsexg.2	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
ceqsexg	|- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Distinct variable group:   x, A

Allowed substitution hints:   ph( x)   ps( x)   V( x)

Proof of Theorem ceqsexg

Step	Hyp	Ref	Expression
1		nfe1 1928	. . 3 |- F/ x E. x ( x = A /\ ph )
2		ceqsexg.1	. . 3 |- F/ x ps
3	1, 2	nfbi 2027	. 2 |- F/ x ( E. x ( x = A /\ ph ) <-> ps )
4		ceqex 3180	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
5		ceqsexg.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
6	4, 5	bibi12d 327	. 2 |- ( x = A -> ( ( ph <-> ph ) <-> ( E. x ( x = A /\ ph ) <-> ps ) ) )
7		biid 244	. 2 |- ( ph <-> ph )
8	3, 6, 7	vtoclg1f 3117	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 189   /\ wa 375   = wceq 1454   E.wex 1673   F/wnf 1677   e. wcel 1897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-ext 2441

This theorem depends on definitions:  df-bi 190  df-an 377  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-clab 2448  df-cleq 2454  df-clel 2457  df-v 3058

This theorem is referenced by:  ceqsexgv  3182