Theorem rspccva 2833

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 2830	. 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 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:  disjne  3468  seex  4320  fconstfvm  5714  fvixp  6681  ordiso2  7012  eqord1  8402  eqord2  8403  seq3caopr2  10438  bccl  10701  2clim  11264  isummulc2  11389  telfsumo2  11430  fsumparts  11433  isumshft  11453  mertenslem2  11499  mertensabs  11500  dvdsprime  12076  mgmlrid  12633  grprinvlem  12639  grpinvex  12718  cnima  13014  dceqnconst  14091  dcapnconst  14092