Theorem csbafv212g 47811

Description: Move class substitution in and out of a function value, analogous to csbfv12 6916, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7444. (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 3858	. . 3 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌(𝐹''''𝐵) = ⦋𝐴 / 𝑥⦌(𝐹''''𝐵))
2		csbeq1 3858	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌𝐹)
3		csbeq1 3858	. . . 4 ⊢ (𝑦 = 𝐴 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐴 / 𝑥⦌𝐵)
4	2, 3	afv2eq12d 47807	. . 3 ⊢ (𝑦 = 𝐴 → (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))
5	1, 4	eqeq12d 2781	. 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 3461	. . 3 ⊢ 𝑦 ∈ V
7		nfcsb1v 3879	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐹
8		nfcsb1v 3879	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
9	7, 8	nfafv2 47810	. . 3 ⊢ Ⅎ𝑥(⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵)
10		csbeq1a 3869	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐹 = ⦋𝑦 / 𝑥⦌𝐹)
11		csbeq1a 3869	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
12	10, 11	afv2eq12d 47807	. . 3 ⊢ (𝑥 = 𝑦 → (𝐹''''𝐵) = (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵))
13	6, 9, 12	csbief 3889	. 2 ⊢ ⦋𝑦 / 𝑥⦌(𝐹''''𝐵) = (⦋𝑦 / 𝑥⦌𝐹''''⦋𝑦 / 𝑥⦌𝐵)
14	5, 13	vtoclg 3525	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌(𝐹''''𝐵) = (⦋𝐴 / 𝑥⦌𝐹''''⦋𝐴 / 𝑥⦌𝐵))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  ⦋csb 3855  ''''cafv2 47800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-iota 6481  df-fun 6527  df-dfat 47711  df-afv2 47801

This theorem is referenced by: (None)