Theorem ectocld 6260

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 6246	. . . 4 |- ( A e. ( B /. R ) -> E. x e. B A = [ x ] R )
2		ectocl.1	. . . 4 |- S = ( B /. R )
3	1, 2	eleq2s 2177	. . 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 2086	. . . . 5 |- ( A = [ x ] R -> ( ph <-> ps ) )
7	4, 6	syl5ibcom 153	. . . 4 |- ( ( ch /\ x e. B ) -> ( A = [ x ] R -> ps ) )
8	7	rexlimdva 2482	. . 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 1285   e. wcel 1434   E.wrex 2354   [cec 6192   /.cqs 6193

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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065

This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-qs 6200

This theorem is referenced by:  ectocl  6261  elqsn0m  6262  qsel  6271