Theorem ectocl 5993

Description: Implicit substitution of class for equivalence class. (Contributed by set.mm contributors, 23-Jul-1995.) (Revised by set.mm contributors, 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 1321	. 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 452	. . 3 |- (( T. /\ x e. B) -> ph)
6	2, 3, 5	ectocld 5992	. 2 |- (( T. /\ A e. S) -> ps)
7	1, 6	mpan 651	1 |- (A e. S -> ps)

Syntax hints:   -> wi 4   <-> wb 176   T. wtru 1316   = wceq 1642   e. wcel 1710  [cec 5946  /.cqs 5947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ral 2620  df-rex 2621  df-v 2862  df-qs 5952

This theorem is referenced by: (None)