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Theorem sbcrot3 26838
Description: Rotate a sequence of three explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
sbcrot3  |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  / 
b ]. [. C  / 
c ]. [. A  / 
a ]. ph )
Distinct variable groups:    A, b    A, c    B, a    C, a   
a, c    a, b
Allowed substitution hints:    ph( a, b, c)    A( a)    B( b, c)    C( b, c)

Proof of Theorem sbcrot3
StepHypRef Expression
1 sbccom 3224 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  / 
b ]. [. A  / 
a ]. [. C  / 
c ]. ph )
2 sbccom 3224 . . 3  |-  ( [. A  /  a ]. [. C  /  c ]. ph  <->  [. C  / 
c ]. [. A  / 
a ]. ph )
32sbcbii 3208 . 2  |-  ( [. B  /  b ]. [. A  /  a ]. [. C  /  c ]. ph  <->  [. B  / 
b ]. [. C  / 
c ]. [. A  / 
a ]. ph )
41, 3bitri 241 1  |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  / 
b ]. [. C  / 
c ]. [. A  / 
a ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [.wsbc 3153
This theorem is referenced by:  sbcrot5  26839  sbcrot3gOLD  26841  sbcrot3OLD  26842  2rexfrabdioph  26847  3rexfrabdioph  26848  4rexfrabdioph  26849  7rexfrabdioph  26851
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-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-v 2950  df-sbc 3154
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