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Theorem sbcco2 2926
Description: A composition law for class substitution. Importantly,  x may occur free in the class expression substituted for  A. (Contributed by NM, 5-Sep-2004.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypothesis
Ref Expression
sbcco2.1  |-  ( x  =  y  ->  A  =  B )
Assertion
Ref Expression
sbcco2  |-  ( [. x  /  y ]. [. B  /  x ]. ph  <->  [. A  /  x ]. ph )
Distinct variable groups:    x, y    ph, y    y, A
Allowed substitution hints:    ph( x)    A( x)    B( x, y)

Proof of Theorem sbcco2
StepHypRef Expression
1 sbsbc 2908 . 2  |-  ( [ x  /  y ]
[. B  /  x ]. ph  <->  [. x  /  y ]. [. B  /  x ]. ph )
2 nfv 1508 . . 3  |-  F/ y
[. A  /  x ]. ph
3 sbcco2.1 . . . . 5  |-  ( x  =  y  ->  A  =  B )
43equcoms 1684 . . . 4  |-  ( y  =  x  ->  A  =  B )
5 dfsbcq 2906 . . . . 5  |-  ( A  =  B  ->  ( [. A  /  x ]. ph  <->  [. B  /  x ]. ph ) )
65bicomd 140 . . . 4  |-  ( A  =  B  ->  ( [. B  /  x ]. ph  <->  [. A  /  x ]. ph ) )
74, 6syl 14 . . 3  |-  ( y  =  x  ->  ( [. B  /  x ]. ph  <->  [. A  /  x ]. ph ) )
82, 7sbie 1764 . 2  |-  ( [ x  /  y ]
[. B  /  x ]. ph  <->  [. A  /  x ]. ph )
91, 8bitr3i 185 1  |-  ( [. x  /  y ]. [. B  /  x ]. ph  <->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331   [wsb 1735   [.wsbc 2904
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-5 1423  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-sbc 2905
This theorem is referenced by: (None)
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