Theorem csbafv12g 47052

Description: Move class substitution in and out of a function value, analogous to csbfv12 6968, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7492. (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 3924	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = ⦋𝐴 / 𝑥⦌(𝐹'''𝐵))
2		csbeq1 3924	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3924	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4	2, 3	afveq12d 47048	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵))
5	1, 4	eqeq12d 2756	. 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 3492	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 3946	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
8		nfcsb1v 3946	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9	7, 8	nfafv 47051	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵)
10		csbeq1a 3935	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
11		csbeq1a 3935	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
12	10, 11	afveq12d 47048	. . 3 ⊢ (𝑥 = 𝑦 → (𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵))
13	6, 9, 12	csbief 3956	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹'''𝐵) = (⦋𝑦 / 𝑥⦌𝐹'''⦋𝑦 / 𝑥⦌𝐵)
14	5, 13	vtoclg 3566	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹'''𝐵) = (⦋𝐴 / 𝑥⦌𝐹'''⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ⦋csb 3921  '''cafv 47032

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-sep 5317  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-ne 2947  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-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-int 4971  df-br 5167  df-opab 5229  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581  df-aiota 47000  df-dfat 47034  df-afv 47035

This theorem is referenced by: (None)