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Theorem csbafv212g 46769
Description: Move class substitution in and out of a function value, analogous to csbfv12 6948, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7466. (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 3894 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹''''𝐵) = 𝐴 / 𝑥(𝐹''''𝐵))
2 csbeq1 3894 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3894 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3afv2eq12d 46765 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
51, 4eqeq12d 2741 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵)))
6 vex 3465 . . 3 𝑦 ∈ V
7 nfcsb1v 3916 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3916 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfafv2 46768 . . 3 𝑥(𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
10 csbeq1a 3905 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3905 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11afv2eq12d 46765 . . 3 (𝑥 = 𝑦 → (𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵))
136, 9, 12csbief 3926 . 2 𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
145, 13vtoclg 3533 1 (𝐴𝑉𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  csb 3891  ''''cafv2 46758
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
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-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4325  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5687  df-rel 5688  df-cnv 5689  df-co 5690  df-dm 5691  df-rn 5692  df-res 5693  df-iota 6505  df-fun 6555  df-dfat 46669  df-afv2 46759
This theorem is referenced by: (None)
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