Theorem ceqsexg 3145

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 1894	. . 3 |- F/ x E. x ( x = A /\ ph )
2		ceqsexg.1	. . 3 |- F/ x ps
3	1, 2	nfbi 1994	. 2 |- F/ x ( E. x ( x = A /\ ph ) <-> ps )
4		ceqex 3144	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
5		ceqsexg.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
6	4, 5	bibi12d 322	. 2 |- ( x = A -> ( ( ph <-> ph ) <-> ( E. x ( x = A /\ ph ) <-> ps ) ) )
7		biid 239	. 2 |- ( ph <-> ph )
8	3, 6, 7	vtoclg1f 3081	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   <-> wb 187   /\ wa 370   = wceq 1437   E.wex 1657   F/wnf 1661   e. wcel 1872

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909  ax-13 2063  ax-ext 2408

This theorem depends on definitions:  df-bi 188  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-v 3024

This theorem is referenced by:  ceqsexgv  3146