Theorem ectocl 4302

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 groups:   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 1538	. 2 |- (A e. S <-> A e. (B/.R))
3		elqsi 4291	. . 3 |- (A e. (B/.R) -> E.x(x e. B /\ A = [x]R))
4		ectocl.2	. . . . . . 7 |- ([x]R = A -> (ph <-> ps))
5	4	eqcoms 1478	. . . . . 6 |- (A = [x]R -> (ph <-> ps))
6		ectocl.3	. . . . . 6 |- (x e. B -> ph)
7	5, 6	syl5bi 208	. . . . 5 |- (A = [x]R -> (x e. B -> ps))
8	7	impcom 351	. . . 4 |- ((x e. B /\ A = [x]R) -> ps)
9	8	19.23aiv 1295	. . 3 |- (E.x(x e. B /\ A = [x]R) -> ps)
10	3, 9	syl 10	. 2 |- (A e. (B/.R) -> ps)
11	2, 10	sylbi 199	1 |- (A e. S -> ps)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E.wex 980  [cec 4259  /.cqs 4260

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  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-rex 1650  df-v 1812  df-qs 4266

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