Theorem ectocld 6603

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 6589	. . . 4 |- ( A e. ( B /. R ) -> E. x e. B A = [ x ] R )
2		ectocl.1	. . . 4 |- S = ( B /. R )
3	1, 2	eleq2s 2272	. . 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 2180	. . . . 5 |- ( A = [ x ] R -> ( ph <-> ps ) )
7	4, 6	syl5ibcom 155	. . . 4 |- ( ( ch /\ x e. B ) -> ( A = [ x ] R -> ps ) )
8	7	rexlimdva 2594	. . 3 |- ( ch -> ( E. x e. B A = [ x ] R -> ps ) )
9	3, 8	syl5 32	. 2 |- ( ch -> ( A e. S -> ps ) )
10	9	imp 124	1 |- ( ( ch /\ A e. S ) -> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1353   e. wcel 2148   E.wrex 2456   [cec 6535   /.cqs 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-qs 6543

This theorem is referenced by:  ectocl  6604  elqsn0m  6605  qsel  6614  eqgen  13091