Theorem ectocl 6770

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 1401	. 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 277	. . 3 |- ( ( T. /\ x e. B ) -> ph )
6	2, 3, 5	ectocld 6769	. 2 |- ( ( T. /\ A e. S ) -> ps )
7	1, 6	mpan 424	1 |- ( A e. S -> ps )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1397   T. wtru 1398   e. wcel 2202   [cec 6699   /.cqs 6700

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-qs 6707

This theorem is referenced by: (None)