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Theorem rspc2v 2735
Description: 2-variable restricted specialization, using implicit substitution. (Contributed by NM, 13-Sep-1999.)
Hypotheses
Ref Expression
rspc2v.1  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
rspc2v.2  |-  ( y  =  B  ->  ( ch 
<->  ps ) )
Assertion
Ref Expression
rspc2v  |-  ( ( 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 rspc2v
StepHypRef Expression
1 nfv 1467 . 2  |-  F/ x ch
2 nfv 1467 . 2  |-  F/ y ps
3 rspc2v.1 . 2  |-  ( x  =  A  ->  ( ph 
<->  ch ) )
4 rspc2v.2 . 2  |-  ( y  =  B  ->  ( ch 
<->  ps ) )
51, 2, 3, 4rspc2 2733 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 1290    e. wcel 1439   A.wral 2360
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ral 2365  df-v 2622
This theorem is referenced by:  rspc2va  2736  rspc3v  2738  disji2  3844  wetriext  4405  f1veqaeq  5562  isorel  5601  fovcl  5764  caovclg  5811  caovcomg  5814  smoel  6079  dcdifsnid  6277  unfiexmid  6682  fiintim  6693  supmoti  6742  supsnti  6754  isotilem  6755  cauappcvgprlem1  7272  caucvgprlemnkj  7279  caucvgprlemnbj  7280  caucvgprprlemval  7301  ltordlem  8014  frecuzrdgrrn  9869  frec2uzrdg  9870  frecuzrdgrcl  9871  frecuzrdgrclt  9876  iseqcaopr3  9964  iseqhomo  9996  seq3homo  9998  climcn2  10752  inopn  11756  basis1  11799  basis2  11800  cncfi  11900
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