Theorem ceqsexg 2931

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		nfcv 2372	. 2 |- F/_ x A
2		nfe1 1542	. . 3 |- F/ x E. x ( x = A /\ ph )
3		ceqsexg.1	. . 3 |- F/ x ps
4	2, 3	nfbi 1635	. 2 |- F/ x ( E. x ( x = A /\ ph ) <-> ps )
5		ceqex 2930	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
6		ceqsexg.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
7	5, 6	bibi12d 235	. 2 |- ( x = A -> ( ( ph <-> ph ) <-> ( E. x ( x = A /\ ph ) <-> ps ) ) )
8		biid 171	. 2 |- ( ph <-> ph )
9	1, 4, 7, 8	vtoclgf 2859	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1395   F/wnf 1506   E.wex 1538   e. wcel 2200

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801

This theorem is referenced by:  ceqsexgv  2932