Theorem rspccv 2904

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 2903	. 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 1395   e. wcel 2200   A.wral 2508

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-v 2801

This theorem is referenced by:  elinti  3932  ofrval  6229  supubti  7166  suplubti  7167  suplocsrlempr  7994  pitonn  8035  peano5uzti  9555  zindd  9565  1arith  12890  basis2  14722  tg2  14734  mopni  15156  metrest  15180  metcnpi  15189  metcnpi2  15190  plycj  15435  decidi  16159  sumdc2  16163