Theorem rspcva 2862

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 2860	. 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 1364   e. wcel 2164   A.wral 2472

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-v 2762

This theorem is referenced by:  supmoti  7052  peano2nnnn  7913  squeeze0  8923  peano2nn  8994  nnsub  9021  zextle  9408  rexuz3  11134  cau3lem  11258  caubnd2  11261  climcn1  11451  dvdsext  11997  mgmidmo  12955  dfgrp3mlem  13170