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Theorem sbccomieg 35101
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 3289 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. ph  ->  A  e.  _V )
2 spesbc 3361 . . 3  |-  ( [. C  /  b ]. [. A  /  a ]. ph  ->  E. b [. A  / 
a ]. ph )
3 sbcex 3289 . . . 4  |-  ( [. A  /  a ]. ph  ->  A  e.  _V )
43exlimiv 1745 . . 3  |-  ( E. b [. A  / 
a ]. ph  ->  A  e.  _V )
52, 4syl 17 . 2  |-  ( [. C  /  b ]. [. A  /  a ]. ph  ->  A  e.  _V )
6 nfcv 2566 . . . 4  |-  F/_ a C
7 nfsbc1v 3299 . . . 4  |-  F/ a
[. A  /  a ]. ph
86, 7nfsbc 3301 . . 3  |-  F/ a
[. C  /  b ]. [. A  /  a ]. ph
9 sbccomieg.1 . . . 4  |-  ( a  =  A  ->  B  =  C )
10 sbceq1a 3290 . . . 4  |-  ( a  =  A  ->  ( ph 
<-> 
[. A  /  a ]. ph ) )
119, 10sbceqbid 3286 . . 3  |-  ( a  =  A  ->  ( [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
128, 11sbciegf 3311 . 2  |-  ( A  e.  _V  ->  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
131, 5, 12pm5.21nii 353 1  |-  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  / 
b ]. [. A  / 
a ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 186    = wceq 1407   E.wex 1635    e. wcel 1844   _Vcvv 3061   [.wsbc 3279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ral 2761  df-rex 2762  df-v 3063  df-sbc 3280
This theorem is referenced by:  2rexfrabdioph  35104  3rexfrabdioph  35105  4rexfrabdioph  35106  6rexfrabdioph  35107  7rexfrabdioph  35108
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