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Theorem csbov2g 5647
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.)
Assertion
Ref Expression
csbov2g  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B F C )  =  ( B F [_ A  /  x ]_ C
) )
Distinct variable groups:    x, B    x, F
Allowed substitution hints:    A( x)    C( x)    V( x)

Proof of Theorem csbov2g
StepHypRef Expression
1 csbov12g 5645 . 2  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C
) )
2 csbconstg 2934 . . 3  |-  ( A  e.  V  ->  [_ A  /  x ]_ B  =  B )
32oveq1d 5628 . 2  |-  ( A  e.  V  ->  ( [_ A  /  x ]_ B F [_ A  /  x ]_ C )  =  ( B F
[_ A  /  x ]_ C ) )
41, 3eqtrd 2117 1  |-  ( A  e.  V  ->  [_ A  /  x ]_ ( B F C )  =  ( B F [_ A  /  x ]_ C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287    e. wcel 1436   [_csb 2922  (class class class)co 5613
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-rex 2361  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-sn 3437  df-pr 3438  df-op 3440  df-uni 3637  df-br 3821  df-iota 4946  df-fv 4989  df-ov 5616
This theorem is referenced by:  csbnegg  7624
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