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Theorem sbcopg 25083
Description: Distribution of class substitution over ordered pairs. (Contributed by Drahflow, 25-Sep-2015.) (Revised by Mario Carneiro, 29-Oct-2015.)
Assertion
Ref Expression
sbcopg  |-  ( 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 sbcopg
StepHypRef Expression
1 nfcsb1v 3247 . . . 4  |-  F/_ x [_ A  /  x ]_ C
2 nfcsb1v 3247 . . . 4  |-  F/_ x [_ A  /  x ]_ D
31, 2nfop 3964 . . 3  |-  F/_ x <. [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >.
43a1i 11 . 2  |-  ( A  e.  _V  ->  F/_ x <. [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >. )
5 csbeq1a 3223 . . 3  |-  ( x  =  A  ->  C  =  [_ A  /  x ]_ C )
6 csbeq1a 3223 . . 3  |-  ( x  =  A  ->  D  =  [_ A  /  x ]_ D )
75, 6opeq12d 3956 . 2  |-  ( x  =  A  ->  <. C ,  D >.  =  <. [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >. )
84, 7csbiegf 3255 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 1649    e. wcel 1721   F/_wnfc 2531   _Vcvv 2920   [_csb 3215   <.cop 3781
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-sn 3784  df-pr 3785  df-op 3787
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