Theorem csbafv12g 46655

Description: Move class substitution in and out of a function value, analogous to csbfv12 6944, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7462. (Contributed by Alexander van der Vekens, 23-Jul-2017.)

Assertion

Ref	Expression
csbafv12g	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵))

Proof of Theorem csbafv12g

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3892	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = ⦋𝐴 / 𝑥⦌(𝐹'''𝐵))
2		csbeq1 3892	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3892	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4	2, 3	afveq12d 46651	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵))
5	1, 4	eqeq12d 2741	. 2 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) ↔ ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵)))
6		vex 3465	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 3914	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
8		nfcsb1v 3914	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9	7, 8	nfafv 46654	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵)
10		csbeq1a 3903	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
11		csbeq1a 3903	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
12	10, 11	afveq12d 46651	. . 3 ⊢ (𝑥 = 𝑦 → (𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵))
13	6, 9, 12	csbief 3924	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵)
14	5, 13	vtoclg 3532	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098  ⦋csb 3889  '''cafv 46635

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-int 4951  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6501  df-fun 6551  df-fv 6557  df-aiota 46603  df-dfat 46637  df-afv 46638

This theorem is referenced by: (None)