Theorem rspccv 2840

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 2839	. 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 1353   e. wcel 2148   A.wral 2455

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741

This theorem is referenced by:  elinti  3855  ofrval  6095  supubti  7000  suplubti  7001  suplocsrlempr  7808  pitonn  7849  peano5uzti  9363  zindd  9373  1arith  12367  basis2  13633  tg2  13645  mopni  14067  metrest  14091  metcnpi  14100  metcnpi2  14101  decidi  14632  sumdc2  14636