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Theorem sbcopg 24694
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 3199 . . . 4  |-  F/_ x [_ A  /  x ]_ C
2 nfcsb1v 3199 . . . 4  |-  F/_ x [_ A  /  x ]_ D
31, 2nfop 3914 . . 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 3175 . . 3  |-  ( x  =  A  ->  C  =  [_ A  /  x ]_ C )
6 csbeq1a 3175 . . 3  |-  ( x  =  A  ->  D  =  [_ A  /  x ]_ D )
75, 6opeq12d 3906 . 2  |-  ( x  =  A  ->  <. C ,  D >.  =  <. [_ A  /  x ]_ C ,  [_ A  /  x ]_ D >. )
84, 7csbiegf 3207 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 1647    e. wcel 1715   F/_wnfc 2489   _Vcvv 2873   [_csb 3167   <.cop 3732
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738
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