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Theorem csbovg 5905
Description: Move class substitution in and out of an operation. (Contributed by NM, 12-Nov-2005.) (Proof shortened by Mario Carneiro, 5-Dec-2016.)
Assertion
Ref Expression
csbovg  |-  ( A  e.  D  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
) )

Proof of Theorem csbovg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3097 . . 3  |-  ( y  =  A  ->  [_ y  /  x ]_ ( B F C )  = 
[_ A  /  x ]_ ( B F C ) )
2 csbeq1 3097 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ F  = 
[_ A  /  x ]_ F )
3 csbeq1 3097 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ B  = 
[_ A  /  x ]_ B )
4 csbeq1 3097 . . . 4  |-  ( y  =  A  ->  [_ y  /  x ]_ C  = 
[_ A  /  x ]_ C )
52, 3, 4oveq123d 5895 . . 3  |-  ( y  =  A  ->  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) )
61, 5eqeq12d 2310 . 2  |-  ( y  =  A  ->  ( [_ y  /  x ]_ ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )  <->  [_ A  /  x ]_ ( B F C )  =  (
[_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C ) ) )
7 vex 2804 . . 3  |-  y  e. 
_V
8 nfcsb1v 3126 . . . 4  |-  F/_ x [_ y  /  x ]_ B
9 nfcsb1v 3126 . . . 4  |-  F/_ x [_ y  /  x ]_ F
10 nfcsb1v 3126 . . . 4  |-  F/_ x [_ y  /  x ]_ C
118, 9, 10nfov 5897 . . 3  |-  F/_ x
( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C )
12 csbeq1a 3102 . . . 4  |-  ( x  =  y  ->  F  =  [_ y  /  x ]_ F )
13 csbeq1a 3102 . . . 4  |-  ( x  =  y  ->  B  =  [_ y  /  x ]_ B )
14 csbeq1a 3102 . . . 4  |-  ( x  =  y  ->  C  =  [_ y  /  x ]_ C )
1512, 13, 14oveq123d 5895 . . 3  |-  ( x  =  y  ->  ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C
) )
167, 11, 15csbief 3135 . 2  |-  [_ y  /  x ]_ ( B F C )  =  ( [_ y  /  x ]_ B [_ y  /  x ]_ F [_ y  /  x ]_ C
)
176, 16vtoclg 2856 1  |-  ( A  e.  D  ->  [_ A  /  x ]_ ( B F C )  =  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ C
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   [_csb 3094  (class class class)co 5874
This theorem is referenced by:  csbov12g  5906
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-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279  df-ov 5877
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