Theorem ceqsexg 2971

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/xps
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 2490	. 2 |- F/_xA
2		nfe1 1732	. . 3 |- F/xE.x(x = A /\ ph)
3		ceqsexg.1	. . 3 |- F/xps
4	2, 3	nfbi 1834	. 2 |- F/x(E.x(x = A /\ ph) <-> ps)
5		ceqex 2970	. . 3 |- (x = A -> (ph <-> E.x(x = A /\ ph)))
6		ceqsexg.2	. . 3 |- (x = A -> (ph <-> ps))
7	5, 6	bibi12d 312	. 2 |- (x = A -> ((ph <-> ph) <-> (E.x(x = A /\ ph) <-> ps)))
8		biid 227	. 2 |- (ph <-> ph)
9	1, 4, 7, 8	vtoclgf 2914	1 |- (A e. V -> (E.x(x = A /\ ph) <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544   = wceq 1642   e. wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  ceqsexgv  2972