Theorem rspccv 2881

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 2-Feb-2006.)

Hypothesis

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

Assertion

Ref	Expression
rspccv	|- ( A. x e. B ph -> ( A e. B -> ps ) )

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

Allowed substitution hint:   ph( x)

Proof of Theorem rspccv

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	com12 30	1 |- ( A. x e. B ph -> ( A e. B -> ps ) )

Syntax hints:   -> wi 4   <-> 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:  elinti  3908  ofrval  6192  supubti  7127  suplubti  7128  suplocsrlempr  7955  pitonn  7996  peano5uzti  9516  zindd  9526  1arith  12805  basis2  14635  tg2  14647  mopni  15069  metrest  15093  metcnpi  15102  metcnpi2  15103  plycj  15348  decidi  15931  sumdc2  15935