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Theorem sbcaltop 24515
Description: Distribution of class substitution over alternate ordered pairs. (Contributed by Scott Fenton, 25-Sep-2015.)
Assertion
Ref Expression
sbcaltop  |-  ( A  e.  _V  ->  [_ A  /  x ]_ << C ,  D >>  =  << [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >> )
Distinct variable group:    x, A
Allowed substitution hints:    C( x)    D( x)

Proof of Theorem sbcaltop
StepHypRef Expression
1 nfcsb1v 3113 . . . 4  |-  F/_ x [_ A  /  x ]_ C
2 nfcsb1v 3113 . . . 4  |-  F/_ x [_ A  /  x ]_ D
31, 2nfaltop 24514 . . 3  |-  F/_ x << [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >>
43a1i 10 . 2  |-  ( A  e.  _V  ->  F/_ x << [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >> )
5 csbeq1a 3089 . . . 4  |-  ( x  =  A  ->  C  =  [_ A  /  x ]_ C )
6 altopeq1 24497 . . . 4  |-  ( C  =  [_ A  /  x ]_ C  ->  << C ,  D >>  =  << [_ A  /  x ]_ C ,  D >> )
75, 6syl 15 . . 3  |-  ( x  =  A  ->  << C ,  D >>  =  << [_ A  /  x ]_ C ,  D >> )
8 csbeq1a 3089 . . . 4  |-  ( x  =  A  ->  D  =  [_ A  /  x ]_ D )
9 altopeq2 24498 . . . 4  |-  ( D  =  [_ A  /  x ]_ D  ->  << [_ A  /  x ]_ C ,  D >>  =  << [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >> )
108, 9syl 15 . . 3  |-  ( x  =  A  ->  << [_ A  /  x ]_ C ,  D >>  =  << [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >> )
117, 10eqtrd 2315 . 2  |-  ( x  =  A  ->  << C ,  D >>  =  << [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >> )
124, 11csbiegf 3121 1  |-  ( A  e.  _V  ->  [_ A  /  x ]_ << C ,  D >>  =  << [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >> )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   F/_wnfc 2406   _Vcvv 2788   [_csb 3081   <<caltop 24490
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-sn 3646  df-pr 3647  df-altop 24492
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