Theorem ectocl 6580

Description: Implicit substitution of class for equivalence class. (Contributed by NM, 23-Jul-1995.) (Revised by Mario Carneiro, 9-Jul-2014.)

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

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

Proof of Theorem ectocl

Step	Hyp	Ref	Expression
1		tru 1352	. 2 |- T.
2		ectocl.1	. . 3 |- S = ( B /. R )
3		ectocl.2	. . 3 |- ( [ x ] R = A -> ( ph <-> ps ) )
4		ectocl.3	. . . 4 |- ( x e. B -> ph )
5	4	adantl 275	. . 3 |- ( ( T. /\ x e. B ) -> ph )
6	2, 3, 5	ectocld 6579	. 2 |- ( ( T. /\ A e. S ) -> ps )
7	1, 6	mpan 422	1 |- ( A e. S -> ps )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   T. wtru 1349   e. wcel 2141   [cec 6511   /.cqs 6512

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-qs 6519

This theorem is referenced by: (None)