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Theorem sbccomieg 26870
Description: Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Hypothesis
Ref Expression
sbccomieg.1  |-  ( a  =  A  ->  B  =  C )
Assertion
Ref Expression
sbccomieg  |-  ( [. 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)

Proof of Theorem sbccomieg
StepHypRef Expression
1 sbcex 3000 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. ph  ->  A  e.  _V )
2 spesbc 3072 . . 3  |-  ( [. C  /  b ]. [. A  /  a ]. ph  ->  E. b [. A  / 
a ]. ph )
3 sbcex 3000 . . . 4  |-  ( [. A  /  a ]. ph  ->  A  e.  _V )
43exlimiv 1666 . . 3  |-  ( E. b [. A  / 
a ]. ph  ->  A  e.  _V )
52, 4syl 15 . 2  |-  ( [. C  /  b ]. [. A  /  a ]. ph  ->  A  e.  _V )
6 nfcv 2419 . . . 4  |-  F/_ a C
7 nfsbc1v 3010 . . . 4  |-  F/ a
[. A  /  a ]. ph
86, 7nfsbc 3012 . . 3  |-  F/ a
[. C  /  b ]. [. A  /  a ]. ph
9 sbccomieg.1 . . . . 5  |-  ( a  =  A  ->  B  =  C )
10 dfsbcq 2993 . . . . 5  |-  ( B  =  C  ->  ( [. B  /  b ]. ph  <->  [. C  /  b ]. ph ) )
119, 10syl 15 . . . 4  |-  ( a  =  A  ->  ( [. B  /  b ]. ph  <->  [. C  /  b ]. ph ) )
12 sbceq1a 3001 . . . . 5  |-  ( a  =  A  ->  ( ph 
<-> 
[. A  /  a ]. ph ) )
1312sbcbidv 3045 . . . 4  |-  ( a  =  A  ->  ( [. C  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
1411, 13bitrd 244 . . 3  |-  ( a  =  A  ->  ( [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
158, 14sbciegf 3022 . 2  |-  ( A  e.  _V  ->  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
161, 5, 15pm5.21nii 342 1  |-  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  / 
b ]. [. A  / 
a ]. ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   E.wex 1528    = wceq 1623    e. wcel 1684   _Vcvv 2788   [.wsbc 2991
This theorem is referenced by:  sbccomiegOLD  26874
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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