Theorem rspc2va 2844

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 2843	. 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 1343   e. wcel 2136   A.wral 2444

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728

This theorem is referenced by:  swopo  4284  ordtri2orexmid  4500  onsucelsucexmid  4507  ordsucunielexmid  4508  ordtri2or2exmid  4548  ontri2orexmidim  4549  isocnv  5779  isotr  5784  ovrspc2v  5868  off  6062  caofrss  6074  oprssdmm  6139  tridc  6865  fidcenumlemrks  6918  seq3caopr2  10417  seq3distr  10448  isprm6  12079  comet  13139  mulcncf  13231  trilpo  13922  neapmkv  13946