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Theorem sbcrot3gOLD 26534
Description: Rotate a sequence of three explicit substitutions, closed theorem. (Contributed by Stefan O'Rear, 11-Oct-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbcrot3gOLD  |-  ( ( A  e.  D  /\  B  e.  E  /\  A. b  C  e.  F
)  ->  ( [. 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)    D( a, b, c)    E( a, b, c)    F( a, b, c)

Proof of Theorem sbcrot3gOLD
StepHypRef Expression
1 sbcrot3 26531 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  / 
b ]. [. C  / 
c ]. [. A  / 
a ]. ph )
21a1i 11 1  |-  ( ( A  e.  D  /\  B  e.  E  /\  A. b  C  e.  F
)  ->  ( [. A  /  a ]. [. B  /  b ]. [. C  /  c ]. ph  <->  [. B  / 
b ]. [. C  / 
c ]. [. A  / 
a ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ w3a 936   A.wal 1546    e. wcel 1717   [.wsbc 3097
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-v 2894  df-sbc 3098
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