Theorem csbfv2g 5522

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 5521	. 2 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( [_ A / x ]_ F ` [_ A / x ]_ B ) )
2		csbconstg 3058	. . 3 |- ( A e. C -> [_ A / x ]_ F = F )
3	2	fveq1d 5487	. 2 |- ( A e. C -> ( [_ A / x ]_ F ` [_ A / x ]_ B ) = ( F ` [_ A / x ]_ B ) )
4	1, 3	eqtrd 2198	1 |- ( A e. C -> [_ A / x ]_ ( F ` B ) = ( F ` [_ A / x ]_ B ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   [_csb 3044   ` cfv 5187

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rex 2449  df-v 2727  df-sbc 2951  df-csb 3045  df-un 3119  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-br 3982  df-iota 5152  df-fv 5195

This theorem is referenced by:  csbfvg  5523  fsumabs  11402  fprodabs  11553