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Theorem csbafv212g 47168
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 7474. (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 3910 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹''''𝐵) = 𝐴 / 𝑥(𝐹''''𝐵))
2 csbeq1 3910 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3910 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3afv2eq12d 47164 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
51, 4eqeq12d 2750 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵)))
6 vex 3481 . . 3 𝑦 ∈ V
7 nfcsb1v 3932 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3932 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfafv2 47167 . . 3 𝑥(𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
10 csbeq1a 3921 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3921 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11afv2eq12d 47164 . . 3 (𝑥 = 𝑦 → (𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵))
136, 9, 12csbief 3942 . 2 𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
145, 13vtoclg 3553 1 (𝐴𝑉𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  csb 3907  ''''cafv2 47157
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-dfat 47068  df-afv2 47158
This theorem is referenced by: (None)
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