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Theorem sbcco3gOLD 3306
Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
sbcco3g.1  |-  ( x  =  A  ->  B  =  C )
Assertion
Ref Expression
sbcco3gOLD  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  / 
y ]. ph ) )
Distinct variable groups:    x, A    ph, x    x, C
Allowed substitution hints:    ph( y)    A( y)    B( x, y)    C( y)    V( x, y)    W( x, y)

Proof of Theorem sbcco3gOLD
StepHypRef Expression
1 sbcco3g.1 . . 3  |-  ( x  =  A  ->  B  =  C )
21sbcco3g 3305 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
32adantr 452 1  |-  ( ( A  e.  V  /\  A. x  B  e.  W
)  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  / 
y ]. ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549    = wceq 1652    e. wcel 1725   [.wsbc 3161
This theorem is referenced by:  fzshftral  11134  2rexfrabdioph  26856  3rexfrabdioph  26857  4rexfrabdioph  26858  6rexfrabdioph  26859  7rexfrabdioph  26860
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 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-sbc 3162  df-csb 3252
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