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Theorem rspc2va 2777
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
StepHypRef Expression
1 rspc2v.1 . . 3  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
2 rspc2v.2 . . 3  |-  ( y  =  B  ->  ( ch 
<->  ps ) )
31, 2rspc2v 2776 . 2  |-  ( ( A  e.  C  /\  B  e.  D )  ->  ( A. x  e.  C  A. y  e.  D  ph  ->  ps ) )
43imp 123 1  |-  ( ( ( A  e.  C  /\  B  e.  D
)  /\  A. x  e.  C  A. y  e.  D  ph )  ->  ps )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    e. wcel 1465   A.wral 2393
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-v 2662
This theorem is referenced by:  swopo  4198  ordtri2orexmid  4408  onsucelsucexmid  4415  ordsucunielexmid  4416  ordtri2or2exmid  4456  isocnv  5680  isotr  5685  off  5962  caofrss  5974  oprssdmm  6037  tridc  6761  fidcenumlemrks  6809  seq3caopr2  10223  seq3distr  10254  isprm6  11752  comet  12595  mulcncf  12687  trilpo  13163
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