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Theorem csbafv212g 42888
Description: Move class substitution in and out of a function value, analogous to csbfv12 6573, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7048. (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 3809 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹''''𝐵) = 𝐴 / 𝑥(𝐹''''𝐵))
2 csbeq1 3809 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3809 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3afv2eq12d 42884 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
51, 4eqeq12d 2808 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵)))
6 vex 3435 . . 3 𝑦 ∈ V
7 nfcsb1v 3828 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3828 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfafv2 42887 . . 3 𝑥(𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
10 csbeq1a 3819 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3819 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11afv2eq12d 42884 . . 3 (𝑥 = 𝑦 → (𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵))
136, 9, 12csbief 3837 . 2 𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
145, 13vtoclg 3505 1 (𝐴𝑉𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1520  wcel 2079  csb 3806  ''''cafv2 42877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-pw 4449  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-iota 6181  df-fun 6219  df-dfat 42788  df-afv2 42878
This theorem is referenced by: (None)
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