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Theorem sbccomiegOLD 26874
Description: Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.)
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 26870 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  / 
b ]. [. A  / 
a ]. ph )
32a1i 10 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 176    /\ wa 358    = wceq 1623    e. wcel 1684   [.wsbc 2991
This theorem is referenced by:  2rexfrabdioph  26877  3rexfrabdioph  26878  4rexfrabdioph  26879  6rexfrabdioph  26880  7rexfrabdioph  26881
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-sbc 2992
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