Theorem ectocl 5361

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 1961	. 2 |- (A e. S <-> A e. (B/.R))
3		elqsi 5349	. . 3 |- (A e. (B/.R) -> E.x e. B A = [x]R)
4		ectocl.2	. . . . . 6 |- ([x]R = A -> (ph <-> ps))
5	4	eqcoms 1887	. . . . 5 |- (A = [x]R -> (ph <-> ps))
6		ectocl.3	. . . . 5 |- (x e. B -> ph)
7	5, 6	syl5cbi 226	. . . 4 |- (x e. B -> (A = [x]R -> ps))
8	7	r19.23aiv 2211	. . 3 |- (E.x e. B A = [x]R -> ps)
9	3, 8	syl 12	. 2 |- (A e. (B/.R) -> ps)
10	2, 9	sylbi 216	1 |- (A e. S -> ps)

Syntax hints:   -> wi 3   <-> wb 163   = wceq 1298   e. wcel 1300  E.wrex 2106  [cec 5316  /.cqs 5317

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ral 2109  df-rex 2110  df-v 2294  df-qs 5323

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