Theorem rspcva 2832

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-2005.)

Hypothesis

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

Assertion

Ref	Expression
rspcva	|- ( ( A e. B /\ A. x e. B ph ) -> ps )

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

Allowed substitution hint:   ph( x)

Proof of Theorem rspcva

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	imp 123	1 |- ( ( A e. B /\ A. x e. B ph ) -> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> 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:  supmoti  6970  peano2nnnn  7815  squeeze0  8820  peano2nn  8890  nnsub  8917  zextle  9303  rexuz3  10954  cau3lem  11078  caubnd2  11081  climcn1  11271  dvdsext  11815  mgmidmo  12626