Theorem rspcva 2882

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 2880	. 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 1373   e. wcel 2178   A.wral 2486

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-v 2778

This theorem is referenced by:  supmoti  7121  peano2nnnn  8001  squeeze0  9012  peano2nn  9083  nnsub  9110  zextle  9499  rexuz3  11416  cau3lem  11540  caubnd2  11543  climcn1  11734  dvdsext  12281  mgmidmo  13319  dfgrp3mlem  13545