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Theorem sbcco3gOLD 3150
Description: Composition of two substitutions. (Contributed by NM, 27-Nov-2005.) (New usage 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 3149 . 2  |-  ( A  e.  V  ->  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  /  y ]. ph ) )
32adantr 451 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 176    /\ wa 358   A.wal 1530    = wceq 1632    e. wcel 1696   [.wsbc 3004
This theorem is referenced by:  fzshftral  10885  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-v 2803  df-sbc 3005  df-csb 3095
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