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Theorem csbafv12g 45845
Description: Move class substitution in and out of a function value, analogous to csbfv12 6940, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7451. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
csbafv12g (𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))

Proof of Theorem csbafv12g
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 csbeq1 3897 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹'''𝐵) = 𝐴 / 𝑥(𝐹'''𝐵))
2 csbeq1 3897 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3897 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3afveq12d 45841 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))
51, 4eqeq12d 2749 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵)))
6 vex 3479 . . 3 𝑦 ∈ V
7 nfcsb1v 3919 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3919 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfafv 45844 . . 3 𝑥(𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵)
10 csbeq1a 3908 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3908 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11afveq12d 45841 . . 3 (𝑥 = 𝑦 → (𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵))
136, 9, 12csbief 3929 . 2 𝑦 / 𝑥(𝐹'''𝐵) = (𝑦 / 𝑥𝐹'''𝑦 / 𝑥𝐵)
145, 13vtoclg 3557 1 (𝐴𝑉𝐴 / 𝑥(𝐹'''𝐵) = (𝐴 / 𝑥𝐹'''𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2107  csb 3894  '''cafv 45825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-res 5689  df-iota 6496  df-fun 6546  df-fv 6552  df-aiota 45793  df-dfat 45827  df-afv 45828
This theorem is referenced by: (None)
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