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Theorem sbcrot5 26746
Description: Rotate a sequence of five explicit substitutions. (Contributed by Stefan O'Rear, 11-Oct-2014.) (Revised by Mario Carneiro, 11-Dec-2016.)
Assertion
Ref Expression
sbcrot5  |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. B  / 
b ]. [. C  / 
c ]. [. D  / 
d ]. [. E  / 
e ]. [. A  / 
a ]. ph )
Distinct variable groups:    A, b    A, c    B, a    C, a   
a, c    a, b    A, d    A, e    D, a    E, a    e, a    a,
d
Allowed substitution hints:    ph( e, a, b, c, d)    A( a)    B( e, b, c, d)    C( e, b, c, d)    D( e, b, c, d)    E( e, b, c, d)

Proof of Theorem sbcrot5
StepHypRef Expression
1 sbcrot3 26745 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. B  / 
b ]. [. C  / 
c ]. [. A  / 
a ]. [. D  / 
d ]. [. E  / 
e ]. ph )
2 sbcrot3 26745 . . . 4  |-  ( [. A  /  a ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. D  / 
d ]. [. E  / 
e ]. [. A  / 
a ]. ph )
32sbcbii 3184 . . 3  |-  ( [. C  /  c ]. [. A  /  a ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. C  / 
c ]. [. D  / 
d ]. [. E  / 
e ]. [. A  / 
a ]. ph )
43sbcbii 3184 . 2  |-  ( [. B  /  b ]. [. C  /  c ]. [. A  /  a ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. B  / 
b ]. [. C  / 
c ]. [. D  / 
d ]. [. E  / 
e ]. [. A  / 
a ]. ph )
51, 4bitri 241 1  |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. [. D  /  d ]. [. E  /  e ]. ph  <->  [. B  / 
b ]. [. C  / 
c ]. [. D  / 
d ]. [. E  / 
e ]. [. A  / 
a ]. ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   [.wsbc 3129
This theorem is referenced by:  sbcrot5OLD  26750  6rexfrabdioph  26757  7rexfrabdioph  26758
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-v 2926  df-sbc 3130
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