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Theorem sbccomiegOLD 26545
Description: Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
sbccomiegOLD.1  |-  ( a  =  A  ->  B  =  C )
Assertion
Ref Expression
sbccomiegOLD  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( [. A  / 
a ]. [. B  / 
b ]. ph  <->  [. C  / 
b ]. [. A  / 
a ]. ph ) )
Distinct variable groups:    A, a,
b    C, a
Allowed substitution hints:    ph( a, b)    B( a, b)    C( b)    V( a, b)    W( a, b)

Proof of Theorem sbccomiegOLD
StepHypRef Expression
1 sbccomiegOLD.1 . . 3  |-  ( a  =  A  ->  B  =  C )
21sbccomieg 26541 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  / 
b ]. [. A  / 
a ]. ph )
32a1i 11 1  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( [. A  / 
a ]. [. B  / 
b ]. ph  <->  [. C  / 
b ]. [. A  / 
a ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1717   [.wsbc 3105
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ral 2655  df-rex 2656  df-v 2902  df-sbc 3106
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