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Theorem sbccomieg 30319
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 3336 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. ph  ->  A  e.  _V )
2 spesbc 3419 . . 3  |-  ( [. C  /  b ]. [. A  /  a ]. ph  ->  E. b [. A  / 
a ]. ph )
3 sbcex 3336 . . . 4  |-  ( [. A  /  a ]. ph  ->  A  e.  _V )
43exlimiv 1693 . . 3  |-  ( E. b [. A  / 
a ]. ph  ->  A  e.  _V )
52, 4syl 16 . 2  |-  ( [. C  /  b ]. [. A  /  a ]. ph  ->  A  e.  _V )
6 nfcv 2624 . . . 4  |-  F/_ a C
7 nfsbc1v 3346 . . . 4  |-  F/ a
[. A  /  a ]. ph
86, 7nfsbc 3348 . . 3  |-  F/ a
[. C  /  b ]. [. A  /  a ]. ph
9 sbccomieg.1 . . . . 5  |-  ( a  =  A  ->  B  =  C )
10 dfsbcq 3328 . . . . 5  |-  ( B  =  C  ->  ( [. B  /  b ]. ph  <->  [. C  /  b ]. ph ) )
119, 10syl 16 . . . 4  |-  ( a  =  A  ->  ( [. B  /  b ]. ph  <->  [. C  /  b ]. ph ) )
12 sbceq1a 3337 . . . . 5  |-  ( a  =  A  ->  ( ph 
<-> 
[. A  /  a ]. ph ) )
1312sbcbidv 3385 . . . 4  |-  ( a  =  A  ->  ( [. C  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
1411, 13bitrd 253 . . 3  |-  ( a  =  A  ->  ( [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
158, 14sbciegf 3358 . 2  |-  ( A  e.  _V  ->  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  /  b ]. [. A  /  a ]. ph ) )
161, 5, 15pm5.21nii 353 1  |-  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  / 
b ]. [. A  / 
a ]. ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1374   E.wex 1591    e. wcel 1762   _Vcvv 3108   [.wsbc 3326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ral 2814  df-rex 2815  df-v 3110  df-sbc 3327
This theorem is referenced by:  2rexfrabdioph  30322  3rexfrabdioph  30323  4rexfrabdioph  30324  6rexfrabdioph  30325  7rexfrabdioph  30326
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