Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  csbafv212g Structured version   Visualization version   GIF version

Theorem csbafv212g 46499
Description: Move class substitution in and out of a function value, analogous to csbfv12 6933, with a direct proof proposed by Mario Carneiro, analogous to csbov123 7447. (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 3891 . . 3 (𝑦 = 𝐴𝑦 / 𝑥(𝐹''''𝐵) = 𝐴 / 𝑥(𝐹''''𝐵))
2 csbeq1 3891 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
3 csbeq1 3891 . . . 4 (𝑦 = 𝐴𝑦 / 𝑥𝐵 = 𝐴 / 𝑥𝐵)
42, 3afv2eq12d 46495 . . 3 (𝑦 = 𝐴 → (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
51, 4eqeq12d 2742 . 2 (𝑦 = 𝐴 → (𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵) ↔ 𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵)))
6 vex 3472 . . 3 𝑦 ∈ V
7 nfcsb1v 3913 . . . 4 𝑥𝑦 / 𝑥𝐹
8 nfcsb1v 3913 . . . 4 𝑥𝑦 / 𝑥𝐵
97, 8nfafv2 46498 . . 3 𝑥(𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
10 csbeq1a 3902 . . . 4 (𝑥 = 𝑦𝐹 = 𝑦 / 𝑥𝐹)
11 csbeq1a 3902 . . . 4 (𝑥 = 𝑦𝐵 = 𝑦 / 𝑥𝐵)
1210, 11afv2eq12d 46495 . . 3 (𝑥 = 𝑦 → (𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵))
136, 9, 12csbief 3923 . 2 𝑦 / 𝑥(𝐹''''𝐵) = (𝑦 / 𝑥𝐹''''𝑦 / 𝑥𝐵)
145, 13vtoclg 3537 1 (𝐴𝑉𝐴 / 𝑥(𝐹''''𝐵) = (𝐴 / 𝑥𝐹''''𝐴 / 𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  csb 3888  ''''cafv2 46488
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 2163  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-iota 6489  df-fun 6539  df-dfat 46399  df-afv2 46489
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator