Theorem cgsexg 1831

Description: Implicit substitution inference for general classes.

Hypotheses

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

Assertion

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

Distinct variable groups:   x,A   ps,x

Proof of Theorem cgsexg

Step	Hyp	Ref	Expression
1		cgsexg.2	. . . 4 |- (ch -> (ph <-> ps))
2	1	biimpa 416	. . 3 |- ((ch /\ ph) -> ps)
3	2	19.23aiv 1295	. 2 |- (E.x(ch /\ ph) -> ps)
4	1	biimprcd 156	. . . . 5 |- (ps -> (ch -> ph))
5	4	ancld 298	. . . 4 |- (ps -> (ch -> (ch /\ ph)))
6	5	19.22dv 1290	. . 3 |- (ps -> (E.xch -> E.x(ch /\ ph)))
7		elex 1819	. . . 4 |- (A e. B -> E.x x = A)
8		cgsexg.1	. . . . 5 |- (x = A -> ch)
9	8	19.22i 1040	. . . 4 |- (E.x x = A -> E.xch)
10	7, 9	syl 10	. . 3 |- (A e. B -> E.xch)
11	6, 10	syl5com 52	. 2 |- (A e. B -> (ps -> E.x(ch /\ ph)))
12	3, 11	impbid2 518	1 |- (A e. B -> (E.x(ch /\ ph) <-> ps))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 963  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812

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