Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sbccomiegOLD Unicode version

Theorem sbccomiegOLD 26977
Description: Commute two explicit substitutions, using an implicit substitution to rewrite the exiting substitution. (Contributed by Stefan O'Rear, 11-Oct-2014.)
Hypothesis
Ref Expression
sbccomiegOLD.1  |-  ( a  =  A  ->  B  =  C )
Assertion
Ref Expression
sbccomiegOLD  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( [. 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)    V( a, b)    W( a, b)

Proof of Theorem sbccomiegOLD
StepHypRef Expression
1 sbccomiegOLD.1 . . 3  |-  ( a  =  A  ->  B  =  C )
21sbccomieg 26973 . 2  |-  ( [. A  /  a ]. [. B  /  b ]. ph  <->  [. C  / 
b ]. [. A  / 
a ]. ph )
32a1i 10 1  |-  ( ( A  e.  V  /\  C  e.  W )  ->  ( [. A  / 
a ]. [. B  / 
b ]. ph  <->  [. C  / 
b ]. [. A  / 
a ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  2rexfrabdioph  26980  3rexfrabdioph  26981  4rexfrabdioph  26982  6rexfrabdioph  26983  7rexfrabdioph  26984
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-sbc 3005
  Copyright terms: Public domain W3C validator