Theorem rspc2va 2714

Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 18-Jun-2014.)

Hypotheses

Ref	Expression
rspc2v.1	|- ( x = A -> ( ph <-> ch ) )
rspc2v.2	|- ( y = B -> ( ch <-> ps ) )

Assertion

Ref	Expression
rspc2va	|- ( ( ( A e. C /\ B e. D ) /\ A. x e. C A. y e. D ph ) -> ps )

Distinct variable groups:   x, y, A   y, B   x, C   x, D, y   ch, x   ps, y

Allowed substitution hints:   ph( x, y)   ps( x)   ch( y)   B( x)   C( y)

Proof of Theorem rspc2va

Step	Hyp	Ref	Expression
1		rspc2v.1	. . 3 |- ( x = A -> ( ph <-> ch ) )
2		rspc2v.2	. . 3 |- ( y = B -> ( ch <-> ps ) )
3	1, 2	rspc2v 2713	. 2 |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )
4	3	imp 122	1 |- ( ( ( A e. C /\ B e. D ) /\ A. x e. C A. y e. D ph ) -> ps )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   = wceq 1284   e. wcel 1433   A.wral 2348

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-v 2603

This theorem is referenced by:  swopo  4061  ordtri2orexmid  4266  onsucelsucexmid  4273  ordsucunielexmid  4274  ordtri2or2exmid  4314  isocnv  5471  isotr  5476  off  5744  caofrss  5755  iseqcaopr2  9461  iseqdistr  9470  isprm6  10526