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Theorem csbfv12g 5551
Description: Move class substitution in and out of a function value. (Contributed by NM, 11-Nov-2005.)
Assertion
Ref Expression
csbfv12g  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )

Proof of Theorem csbfv12g
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 csbiotag 5209 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [. A  /  x ]. B F y ) )
2 sbcbrg 4057 . . . . 5  |-  ( A  e.  C  ->  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y
) )
3 csbconstg 3071 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ y  =  y )
43breq2d 4015 . . . . 5  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
52, 4bitrd 188 . . . 4  |-  ( A  e.  C  ->  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
65iotabidv 5199 . . 3  |-  ( A  e.  C  ->  ( iota y [. A  /  x ]. B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
71, 6eqtrd 2210 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
8 df-fv 5224 . . 3  |-  ( F `
 B )  =  ( iota y B F y )
98csbeq2i 3084 . 2  |-  [_ A  /  x ]_ ( F `
 B )  = 
[_ A  /  x ]_ ( iota y B F y )
10 df-fv 5224 . 2  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F
y )
117, 9, 103eqtr4g 2235 1  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( F `
 B )  =  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1353    e. wcel 2148   [.wsbc 2962   [_csb 3057   class class class wbr 4003   iotacio 5176   ` cfv 5216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-iota 5178  df-fv 5224
This theorem is referenced by:  csbfv2g  5552
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