Theorem rspcva 2921

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)

Hypothesis

Ref	Expression
rspcv.1	|- ( x = A -> ( ph <-> ps ) )

Assertion

Ref	Expression
rspcva	|- ( ( A e. B /\ A. x e. B ph ) -> ps )

Distinct variable groups:   x, A   x, B   ps, x

Allowed substitution hint:   ph( x)

Proof of Theorem rspcva

Step	Hyp	Ref	Expression
1		rspcv.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
2	1	rspcv 2919	. 2 |- ( A e. B -> ( A. x e. B ph -> ps ) )
3	2	imp 124	1 |- ( ( A e. B /\ A. x e. B ph ) -> ps )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   = wceq 1398   e. wcel 2205   A.wral 2522

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-v 2817

This theorem is referenced by:  supmoti  7286  peano2nnnn  8170  squeeze0  9180  peano2nn  9251  nnsub  9278  zextle  9672  rexuz3  11679  cau3lem  11803  caubnd2  11806  climcn1  11997  dvdsext  12545  mgmidmo  13602  dfgrp3mlem  13828