Theorem rspccv 2920

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 2919	. 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 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:  elinti  3960  ofrval  6279  supubti  7292  suplubti  7293  suplocsrlempr  8124  pitonn  8165  peano5uzti  9689  zindd  9699  1arith  13069  basis2  14930  tg2  14942  mopni  15364  metrest  15388  metcnpi  15397  metcnpi2  15398  plycj  15643  eupthseg  16464  decidi  16584  sumdc2  16588