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Theorem sbcco2 3176
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 3157 . 2  |-  ( [ x  /  y ]
[. B  /  x ]. ph  <->  [. x  /  y ]. [. B  /  x ]. ph )
2 nfv 1629 . . 3  |-  F/ y
[. A  /  x ]. ph
3 sbcco2.1 . . . . 5  |-  ( x  =  y  ->  A  =  B )
43equcoms 1693 . . . 4  |-  ( y  =  x  ->  A  =  B )
5 dfsbcq 3155 . . . . 5  |-  ( A  =  B  ->  ( [. A  /  x ]. ph  <->  [. B  /  x ]. ph ) )
65bicomd 193 . . . 4  |-  ( A  =  B  ->  ( [. B  /  x ]. ph  <->  [. A  /  x ]. ph ) )
74, 6syl 16 . . 3  |-  ( y  =  x  ->  ( [. B  /  x ]. ph  <->  [. A  /  x ]. ph ) )
82, 7sbie 2122 . 2  |-  ( [ x  /  y ]
[. B  /  x ]. ph  <->  [. A  /  x ]. ph )
91, 8bitr3i 243 1  |-  ( [. x  /  y ]. [. B  /  x ]. ph  <->  [. A  /  x ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1652   [wsb 1658   [.wsbc 3153
This theorem is referenced by:  tfinds2  4835
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-sbc 3154
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