Theorem rspccv 2831

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 2830	. 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 104   = wceq 1348   e. wcel 2141   A.wral 2448

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732

This theorem is referenced by:  elinti  3840  ofrval  6071  supubti  6976  suplubti  6977  suplocsrlempr  7769  pitonn  7810  peano5uzti  9320  zindd  9330  1arith  12319  basis2  12840  tg2  12854  mopni  13276  metrest  13300  metcnpi  13309  metcnpi2  13310  decidi  13830  sumdc2  13834