Theorem ceqsexg 2854

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 2308	. 2 |- F/_ x A
2		nfe1 1484	. . 3 |- F/ x E. x ( x = A /\ ph )
3		ceqsexg.1	. . 3 |- F/ x ps
4	2, 3	nfbi 1577	. 2 |- F/ x ( E. x ( x = A /\ ph ) <-> ps )
5		ceqex 2853	. . 3 |- ( x = A -> ( ph <-> E. x ( x = A /\ ph ) ) )
6		ceqsexg.2	. . 3 |- ( x = A -> ( ph <-> ps ) )
7	5, 6	bibi12d 234	. 2 |- ( x = A -> ( ( ph <-> ph ) <-> ( E. x ( x = A /\ ph ) <-> ps ) ) )
8		biid 170	. 2 |- ( ph <-> ph )
9	1, 4, 7, 8	vtoclgf 2784	1 |- ( A e. V -> ( E. x ( x = A /\ ph ) <-> ps ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   F/wnf 1448   E.wex 1480   e. wcel 2136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728

This theorem is referenced by:  ceqsexgv  2855