Theorem rspc2va 2848

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 2847	. 2 |- ( ( A e. C /\ B e. D ) -> ( A. x e. C A. y e. D ph -> ps ) )
4	3	imp 123	1 |- ( ( ( A e. C /\ B e. D ) /\ A. x e. C A. y e. D 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:  swopo  4291  ordtri2orexmid  4507  onsucelsucexmid  4514  ordsucunielexmid  4515  ordtri2or2exmid  4555  ontri2orexmidim  4556  isocnv  5790  isotr  5795  ovrspc2v  5879  off  6073  caofrss  6085  oprssdmm  6150  tridc  6877  fidcenumlemrks  6930  seq3caopr2  10438  seq3distr  10469  isprm6  12101  mhmpropd  12689  comet  13293  mulcncf  13385  trilpo  14075  neapmkv  14099