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Theorem sbequ 1812
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 1811 . 2  |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) )
2 sbequi 1811 . . 3  |-  ( y  =  x  ->  ( [ y  /  z ] ph  ->  [ x  /  z ] ph ) )
32equcoms 1684 . 2  |-  ( x  =  y  ->  ( [ y  /  z ] ph  ->  [ x  /  z ] ph ) )
41, 3impbid 128 1  |-  ( x  =  y  ->  ( [ x  /  z ] ph  <->  [ y  /  z ] ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104   [wsb 1735
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736
This theorem is referenced by:  drsb2  1813  sbco2vlem  1917  sbco2v  1921  sbco2yz  1936  sbcocom  1943  sb10f  1970  hbsb4  1987  nfsb4or  1998  sb8eu  2012  sb8euh  2022  cbvab  2263  cbvralf  2648  cbvrexf  2649  cbvreu  2652  cbvralsv  2668  cbvrexsv  2669  cbvrab  2684  cbvreucsf  3064  cbvrabcsf  3065  sbss  3471  disjiun  3924  cbvopab1  4001  cbvmpt  4023  tfis  4497  findes  4517  cbviota  5093  sb8iota  5095  cbvriota  5740  uzind4s  9392  bezoutlemmain  11693  cbvrald  13009  setindft  13177
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