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Theorem csbafv212g 47231
Description: Move class substitution in and out of a function value, analogous to csbfv12 6954, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7475. (Contributed by AV, 4-Sep-2022.)
Assertion
Ref Expression
csbafv212g (𝐴𝑉𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))

Proof of Theorem csbafv212g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3902 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹''''𝐵) = 𝐴 / 𝑥(𝐹''''𝐵))
2 csbeq1 3902 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3902 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3afv2eq12d 47227 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
51, 4eqeq12d 2753 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵)))
6 vex 3484 . . 3 𝑦 ∈ V
7 nfcsb1v 3923 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3923 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfafv2 47230 . . 3 𝑥(𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
10 csbeq1a 3913 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3913 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11afv2eq12d 47227 . . 3 (𝑥 = 𝑦 → (𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵))
136, 9, 12csbief 3933 . 2 𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
145, 13vtoclg 3554 1 (𝐴𝑉𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  csb 3899  ''''cafv2 47220
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-dfat 47131  df-afv2 47221
This theorem is referenced by: (None)
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