Theorem ectocld 6195

Description: Implicit substitution of class for equivalence class. (Contributed by Mario Carneiro, 9-Jul-2014.)

Hypotheses

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

Assertion

Ref	Expression
ectocld	|- ( ( ch /\ A e. S ) -> ps )

Distinct variable groups:   x, A   x, B   x, R   ps, x   ch, x

Allowed substitution hints:   ph( x)   S( x)

Proof of Theorem ectocld

Step	Hyp	Ref	Expression
1		elqsi 6181	. . . 4 |- ( A e. ( B /. R ) -> E. x e. B A = [ x ] R )
2		ectocl.1	. . . 4 |- S = ( B /. R )
3	1, 2	eleq2s 2173	. . 3 |- ( A e. S -> E. x e. B A = [ x ] R )
4		ectocld.3	. . . . 5 |- ( ( ch /\ x e. B ) -> ph )
5		ectocl.2	. . . . . 6 |- ( [ x ] R = A -> ( ph <-> ps ) )
6	5	eqcoms 2084	. . . . 5 |- ( A = [ x ] R -> ( ph <-> ps ) )
7	4, 6	syl5ibcom 153	. . . 4 |- ( ( ch /\ x e. B ) -> ( A = [ x ] R -> ps ) )
8	7	rexlimdva 2477	. . 3 |- ( ch -> ( E. x e. B A = [ x ] R -> ps ) )
9	3, 8	syl5 32	. 2 |- ( ch -> ( A e. S -> ps ) )
10	9	imp 122	1 |- ( ( ch /\ A e. S ) -> ps )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   E.wrex 2349   [cec 6127   /.cqs 6128

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-qs 6135

This theorem is referenced by:  ectocl  6196  elqsn0m  6197  qsel  6206