Theorem ectocld 6567

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 6553	. . . 4 |- ( A e. ( B /. R ) -> E. x e. B A = [ x ] R )
2		ectocl.1	. . . 4 |- S = ( B /. R )
3	1, 2	eleq2s 2261	. . 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 2168	. . . . 5 |- ( A = [ x ] R -> ( ph <-> ps ) )
7	4, 6	syl5ibcom 154	. . . 4 |- ( ( ch /\ x e. B ) -> ( A = [ x ] R -> ps ) )
8	7	rexlimdva 2583	. . 3 |- ( ch -> ( E. x e. B A = [ x ] R -> ps ) )
9	3, 8	syl5 32	. 2 |- ( ch -> ( A e. S -> ps ) )
10	9	imp 123	1 |- ( ( ch /\ A e. S ) -> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1343   e. wcel 2136   E.wrex 2445   [cec 6499   /.cqs 6500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-qs 6507

This theorem is referenced by:  ectocl  6568  elqsn0m  6569  qsel  6578