Theorem rspccva 2759

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)

Hypothesis

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

Assertion

Ref	Expression
rspccva	|- ( ( 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 rspccva

Step	Hyp	Ref	Expression
1		rspcv.1	. . 3 |- ( x = A -> ( ph <-> ps ) )
2	1	rspcv 2756	. 2 |- ( A e. B -> ( A. x e. B ph -> ps ) )
3	2	impcom 124	1 |- ( ( A. x e. B ph /\ A e. B ) -> ps )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   = wceq 1314   e. wcel 1463   A.wral 2390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-v 2659

This theorem is referenced by:  disjne  3382  seex  4217  fconstfvm  5592  grprinvlem  5919  fvixp  6551  ordiso2  6872  eqord1  8164  eqord2  8165  seq3caopr2  10148  bccl  10406  2clim  10962  isummulc2  11087  telfsumo2  11128  fsumparts  11131  isumshft  11151  mertenslem2  11197  mertensabs  11198  dvdsprime  11649  cnima  12231