Theorem rspcva 2814

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 2812	. 2 |- ( A e. B -> ( A. x e. B ph -> ps ) )
3	2	imp 123	1 |- ( ( A e. B /\ A. x e. B ph ) -> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1335   e. wcel 2128   A.wral 2435

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-v 2714

This theorem is referenced by:  supmoti  6937  peano2nnnn  7773  squeeze0  8775  peano2nn  8845  nnsub  8872  zextle  9255  rexuz3  10890  cau3lem  11014  caubnd2  11017  climcn1  11205  dvdsext  11746