Theorem csbfv2g 6886

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

Assertion

Ref	Expression
csbfv2g	⊢ (𝐴 ∈ 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		csbfv12 6885	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)
2		csbconstg 3856	. . 3 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌𝐹 = 𝐹)
3	2	fveq1d 6842	. 2 ⊢ (𝐴 ∈ 𝐶 → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵))
4	1, 3	eqtrid 2783	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ⦋csb 3837  ‘cfv 6498

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-dm 5641  df-iota 6454  df-fv 6506

This theorem is referenced by:  csbfv  6887  ixpsnval  8848  swrdspsleq  14628  sumeq2ii  15655  fsumabs  15764  prodeq2ii  15876  fprodabs  15939  ixpsnbasval  21203  coe1fzgsumdlem  22268  evl1gsumdlem  22321  pm2mp  22790  cayhamlem4  22853  iuninc  32630  cdlemk39s  41385  evl1gprodd  42556  minregex  43961