Theorem csbfv2g 5210

Description: Move class substitution in and out of a function value. (Contributed by NM, 10-Nov-2005.)

Assertion

Ref	Expression
csbfv2g	|- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( F ` [_ A / x ]_ B ) )

Distinct variable group:   x, F

Allowed substitution hints:   A( x)   B( x)   C( x)

Proof of Theorem csbfv2g

Step	Hyp	Ref	Expression
1		csbfv12g 5209	. 2 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )
2		csbconstg 2864	. . 3 |- ( A e. C -> [_ A / x ]_ F = F )
3	2	fveq1d 5180	. 2 |- ( A e. C -> ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( F ` [_ A / x ]_ B ) )
4	1, 3	eqtrd 2072	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   [_csb 2852   ` cfv 4902

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-iota 4867  df-fv 4910

This theorem is referenced by:  csbfvg  5211