Theorem ectocl 3238

Description: Implicit substitution of class for equivalence class.

Hypotheses

Ref	Expression
ectocl.1	|- S = (B/.R)
ectocl.2	|- ([x]R = A -> (ph <-> ps))
ectocl.3	|- (x e. B -> ph)

Assertion

Ref	Expression
ectocl	|- (A e. S -> ps)

Distinct variable group(s):   x,A   x,B   x,R   ps,x

Proof of Theorem ectocl

Step	Hyp	Ref	Expression
1		ectocl.1	. . 3 |- S = (B/.R)
2	1	eleq2i 1153	. 2 |- (A e. S <-> A e. (B/.R))
3		elqsi 3228	. . 3 |- (A e. (B/.R) -> E.x(x e. B /\ A = [x]R))
4		ectocl.2	. . . . . . . 8 |- ([x]R = A -> (ph <-> ps))
5	4	cleqcoms 1104	. . . . . . 7 |- (A = [x]R -> (ph <-> ps))
6		ectocl.3	. . . . . . 7 |- (x e. B -> ph)
7	5, 6	syl5bi 183	. . . . . 6 |- (A = [x]R -> (x e. B -> ps))
8	7	com12 13	. . . . 5 |- (x e. B -> (A = [x]R -> ps))
9	8	imp 277	. . . 4 |- ((x e. B /\ A = [x]R) -> ps)
10	9	19.23aiv 952	. . 3 |- (E.x(x e. B /\ A = [x]R) -> ps)
11	3, 10	syl 12	. 2 |- (A e. (B/.R) -> ps)
12	2, 11	sylbi 174	1 |- (A e. S -> ps)

Syntax hints:   -> wi 2   <-> wb 127   /\ wa 196  E.wex 678   = wceq 1091   e. wcel 1092  [cec 3198  /.cqs 3199

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-qs 3205

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