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Theorem sbccomiegOLD 26844
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 26840 . 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 1652    e. wcel 1725   [.wsbc 3153
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-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-sbc 3154
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