Theorem cgsexg 2890

Description: Implicit substitution inference for general classes. (Contributed by NM, 26-Aug-2007.)

Hypotheses

Ref	Expression
cgsexg.1	|- (x = A -> ch)
cgsexg.2	|- (ch -> (ph <-> ps))

Assertion

Ref	Expression
cgsexg	|- (A e. V -> (E.x(ch /\ ph) <-> ps))

Distinct variable groups:   x,A   ps,x

Allowed substitution hints:   ph(x)   ch(x)   V(x)

Proof of Theorem cgsexg

Step	Hyp	Ref	Expression
1		cgsexg.2	. . . 4 |- (ch -> (ph <-> ps))
2	1	biimpa 470	. . 3 |- ((ch /\ ph) -> ps)
3	2	exlimiv 1634	. 2 |- (E.x(ch /\ ph) -> ps)
4		elisset 2869	. . . 4 |- (A e. V -> E.x x = A)
5		cgsexg.1	. . . . 5 |- (x = A -> ch)
6	5	eximi 1576	. . . 4 |- (E.x x = A -> E.xch)
7	4, 6	syl 15	. . 3 |- (A e. V -> E.xch)
8	1	biimprcd 216	. . . . 5 |- (ps -> (ch -> ph))
9	8	ancld 536	. . . 4 |- (ps -> (ch -> (ch /\ ph)))
10	9	eximdv 1622	. . 3 |- (ps -> (E.xch -> E.x(ch /\ ph)))
11	7, 10	syl5com 26	. 2 |- (A e. V -> (ps -> E.x(ch /\ ph)))
12	3, 11	impbid2 195	1 |- (A e. V -> (E.x(ch /\ ph) <-> ps))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  E.wex 1541   = 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-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861

This theorem is referenced by: (None)