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Theorem csbfv12g 5423
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 5084 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [. A  /  x ]. B F y ) )
2 sbcbrg 3950 . . . . 5  |-  ( A  e.  C  ->  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y
) )
3 csbconstg 2985 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ y  =  y )
43breq2d 3909 . . . . 5  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
52, 4bitrd 187 . . . 4  |-  ( A  e.  C  ->  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
65iotabidv 5077 . . 3  |-  ( A  e.  C  ->  ( iota y [. A  /  x ]. B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
71, 6eqtrd 2148 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
8 df-fv 5099 . . 3  |-  ( F `
 B )  =  ( iota y B F y )
98csbeq2i 2997 . 2  |-  [_ A  /  x ]_ ( F `
 B )  = 
[_ A  /  x ]_ ( iota y B F y )
10 df-fv 5099 . 2  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F
y )
117, 9, 103eqtr4g 2173 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 1314    e. wcel 1463   [.wsbc 2880   [_csb 2973   class class class wbr 3897   iotacio 5054   ` cfv 5091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-rex 2397  df-v 2660  df-sbc 2881  df-csb 2974  df-un 3043  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-iota 5056  df-fv 5099
This theorem is referenced by:  csbfv2g  5424
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