Theorem csbfv2g 6969

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 6968	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)
2		csbconstg 3940	. . 3 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌𝐹 = 𝐹)
3	2	fveq1d 6922	. 2 ⊢ (𝐴 ∈ 𝐶 → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵))
4	1, 3	eqtrid 2792	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ⦋csb 3921  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-dm 5710  df-iota 6525  df-fv 6581

This theorem is referenced by:  csbfv  6970  ixpsnval  8958  swrdspsleq  14713  sumeq2ii  15741  fsumabs  15849  prodeq2ii  15959  fprodabs  16022  ixpsnbasval  21238  coe1fzgsumdlem  22328  evl1gsumdlem  22381  pm2mp  22852  cayhamlem4  22915  iuninc  32583  cdlemk39s  40896  evl1gprodd  42074  minregex  43496