Theorem csbafv212g 47224

Description: Move class substitution in and out of a function value, analogous to csbfv12 6909, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7434. (Contributed by AV, 4-Sep-2022.)

Assertion

Ref	Expression
csbafv212g	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))

Proof of Theorem csbafv212g

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3868	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹''''𝐵) = ⦋𝐴 / 𝑥⦌(𝐹''''𝐵))
2		csbeq1 3868	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3868	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4	2, 3	afv2eq12d 47220	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))
5	1, 4	eqeq12d 2746	. 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 3454	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 3889	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
8		nfcsb1v 3889	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9	7, 8	nfafv2 47223	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵)
10		csbeq1a 3879	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
11		csbeq1a 3879	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
12	10, 11	afv2eq12d 47220	. . 3 ⊢ (𝑥 = 𝑦 → (𝐹''''𝐵) = (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵))
13	6, 9, 12	csbief 3899	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹''''𝐵) = (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵)
14	5, 13	vtoclg 3523	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ⦋csb 3865  ''''cafv2 47213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-dfat 47124  df-afv2 47214

This theorem is referenced by: (None)