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Theorem vtocl3gf 3260
Description: Implicit substitution of a class for a setvar variable. (Contributed by NM, 10-Aug-2013.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
vtocl3gf.a 𝑥𝐴
vtocl3gf.b 𝑦𝐴
vtocl3gf.c 𝑧𝐴
vtocl3gf.d 𝑦𝐵
vtocl3gf.e 𝑧𝐵
vtocl3gf.f 𝑧𝐶
vtocl3gf.1 𝑥𝜓
vtocl3gf.2 𝑦𝜒
vtocl3gf.3 𝑧𝜃
vtocl3gf.4 (𝑥 = 𝐴 → (𝜑𝜓))
vtocl3gf.5 (𝑦 = 𝐵 → (𝜓𝜒))
vtocl3gf.6 (𝑧 = 𝐶 → (𝜒𝜃))
vtocl3gf.7 𝜑
Assertion
Ref Expression
vtocl3gf ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝜃)

Proof of Theorem vtocl3gf
StepHypRef Expression
1 elex 3203 . . 3 (𝐴𝑉𝐴 ∈ V)
2 vtocl3gf.d . . . 4 𝑦𝐵
3 vtocl3gf.e . . . 4 𝑧𝐵
4 vtocl3gf.f . . . 4 𝑧𝐶
5 vtocl3gf.b . . . . . 6 𝑦𝐴
65nfel1 2781 . . . . 5 𝑦 𝐴 ∈ V
7 vtocl3gf.2 . . . . 5 𝑦𝜒
86, 7nfim 1827 . . . 4 𝑦(𝐴 ∈ V → 𝜒)
9 vtocl3gf.c . . . . . 6 𝑧𝐴
109nfel1 2781 . . . . 5 𝑧 𝐴 ∈ V
11 vtocl3gf.3 . . . . 5 𝑧𝜃
1210, 11nfim 1827 . . . 4 𝑧(𝐴 ∈ V → 𝜃)
13 vtocl3gf.5 . . . . 5 (𝑦 = 𝐵 → (𝜓𝜒))
1413imbi2d 330 . . . 4 (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
15 vtocl3gf.6 . . . . 5 (𝑧 = 𝐶 → (𝜒𝜃))
1615imbi2d 330 . . . 4 (𝑧 = 𝐶 → ((𝐴 ∈ V → 𝜒) ↔ (𝐴 ∈ V → 𝜃)))
17 vtocl3gf.a . . . . 5 𝑥𝐴
18 vtocl3gf.1 . . . . 5 𝑥𝜓
19 vtocl3gf.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
20 vtocl3gf.7 . . . . 5 𝜑
2117, 18, 19, 20vtoclgf 3255 . . . 4 (𝐴 ∈ V → 𝜓)
222, 3, 4, 8, 12, 14, 16, 21vtocl2gf 3259 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐴 ∈ V → 𝜃))
231, 22mpan9 486 . 2 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → 𝜃)
24233impb 1257 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝜃)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wnf 1705  wcel 1992  wnfc 2754  Vcvv 3191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-v 3193
This theorem is referenced by:  vtocl3gaf  3266
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