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Theorem sbequ 1763
Description: An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbequ  |-  ( x  =  y  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )

Proof of Theorem sbequ
StepHypRef Expression
1 sbequi 1762 . 2  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
2 sbequi 1762 . . 3  |-  ( y  =  x  ->  ( [ y  /  z ] ph  ->  [ x  /  z ] ph ) )
32equcoms 1636 . 2  |-  ( x  =  y  ->  ( [ y  /  z ] ph  ->  [ x  /  z ] ph ) )
41, 3impbid 127 1  |-  ( x  =  y  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   [wsb 1687
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1688
This theorem is referenced by:  drsb2  1764  sbco2vlem  1863  sbco2yz  1880  sbcocom  1887  sb10f  1914  hbsb4  1931  nfsb4or  1942  sb8eu  1956  sb8euh  1966  cbvab  2205  cbvralf  2576  cbvrexf  2577  cbvreu  2580  cbvralsv  2593  cbvrexsv  2594  cbvrab  2608  cbvreucsf  2975  cbvrabcsf  2976  sbss  3366  cbvopab1  3871  cbvmpt  3892  tfis  4352  findes  4372  cbviota  4922  sb8iota  4924  cbvriota  5529  uzind4s  8811  bezoutlemmain  10594  cbvrald  10858  setindft  11027
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