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Theorem csbfv12g 5730
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 5439 . . 3  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [. A  /  x ]. B F y ) )
2 sbcbrg 4253 . . . . 5  |-  ( A  e.  C  ->  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y
) )
3 csbconstg 3257 . . . . . 6  |-  ( A  e.  C  ->  [_ A  /  x ]_ y  =  y )
43breq2d 4216 . . . . 5  |-  ( A  e.  C  ->  ( [_ A  /  x ]_ B [_ A  /  x ]_ F [_ A  /  x ]_ y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
52, 4bitrd 245 . . . 4  |-  ( A  e.  C  ->  ( [. A  /  x ]. B F y  <->  [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
65iotabidv 5431 . . 3  |-  ( A  e.  C  ->  ( iota y [. A  /  x ]. B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
71, 6eqtrd 2467 . 2  |-  ( A  e.  C  ->  [_ A  /  x ]_ ( iota y B F y )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F y ) )
8 df-fv 5454 . . 3  |-  ( F `
 B )  =  ( iota y B F y )
98csbeq2i 3269 . 2  |-  [_ A  /  x ]_ ( F `
 B )  = 
[_ A  /  x ]_ ( iota y B F y )
10 df-fv 5454 . 2  |-  ( [_ A  /  x ]_ F `  [_ A  /  x ]_ B )  =  ( iota y [_ A  /  x ]_ B [_ A  /  x ]_ F
y )
117, 9, 103eqtr4g 2492 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 1652    e. wcel 1725   [.wsbc 3153   [_csb 3243   class class class wbr 4204   iotacio 5408   ` cfv 5446
This theorem is referenced by:  csbfv2g  5732  cdlemk42  31675
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-iota 5410  df-fv 5454
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