Theorem rspccva 2783

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 2780	. 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 1331   e. wcel 1480   A.wral 2414

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683

This theorem is referenced by:  disjne  3411  seex  4252  fconstfvm  5631  grprinvlem  5958  fvixp  6590  ordiso2  6913  eqord1  8238  eqord2  8239  seq3caopr2  10248  bccl  10506  2clim  11063  isummulc2  11188  telfsumo2  11229  fsumparts  11232  isumshft  11252  mertenslem2  11298  mertensabs  11299  dvdsprime  11792  cnima  12378