Theorem ceqsexg 3231

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 1841	. . 3 |- F/ x E. x ( x = A /\ ph )
2		ceqsexg.1	. . 3 |- F/ x ps
3	1, 2	nfbi 1935	. 2 |- F/ x ( E. x ( x = A /\ ph ) <-> ps )
4		ceqex 3230	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
5		ceqsexg.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
6	4, 5	bibi12d 321	. 2 |- ( x = A -> ( ( ph <-> ph ) <-> ( E. x ( x = A /\ ph ) <-> ps ) ) )
7		biid 236	. 2 |- ( ph <-> ph )
8	3, 6, 7	vtoclg1f 3166	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   = wceq 1395   E.wex 1613   F/wnf 1617   e. wcel 1819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-12 1855  ax-13 2000  ax-ext 2435

This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-v 3111

This theorem is referenced by:  ceqsexgv  3232