Theorem csbfv2g 6931

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 6930	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵)
2		csbconstg 3866	. . 3 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌𝐹 = 𝐹)
3	2	fveq1d 6887	. 2 ⊢ (𝐴 ∈ 𝐶 → (⦋𝐴 / 𝑥⦌𝐹‘⦋𝐴 / 𝑥⦌𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵))
4	1, 3	eqtrid 2808	1 ⊢ (𝐴 ∈ 𝐶 → ⦋𝐴 / 𝑥⦌(𝐹‘𝐵) = (𝐹‘⦋𝐴 / 𝑥⦌𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ⦋csb 3847  ‘cfv 6538

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-dm 5661  df-iota 6494  df-fv 6546

This theorem is used by:  csbfv  6932  ixpsnval  8928  swrdspsleq  14815  sumeq2ii  15860  fsumabs  15968  prodeq2ii  16080  fprodabs  16141  ixpsnbasval  21483  coe1fzgsumdlem  22621  evl1gsumdlem  22674  pm2mp  23143  cayhamlem4  23206  iuninc  33155  cdlemk39s  41996  evl1gprodd  43167  minregex  44534