ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  vtocl3gf GIF version

Theorem vtocl3gf 2682
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 2630 . . 3 (𝐴𝑉𝐴 ∈ V)
2 vtocl3gf.d . . . 4 𝑦𝐵
3 vtocl3gf.e . . . 4 𝑧𝐵
4 vtocl3gf.f . . . 4 𝑧𝐶
5 vtocl3gf.b . . . . . 6 𝑦𝐴
65nfel1 2239 . . . . 5 𝑦 𝐴 ∈ V
7 vtocl3gf.2 . . . . 5 𝑦𝜒
86, 7nfim 1509 . . . 4 𝑦(𝐴 ∈ V → 𝜒)
9 vtocl3gf.c . . . . . 6 𝑧𝐴
109nfel1 2239 . . . . 5 𝑧 𝐴 ∈ V
11 vtocl3gf.3 . . . . 5 𝑧𝜃
1210, 11nfim 1509 . . . 4 𝑧(𝐴 ∈ V → 𝜃)
13 vtocl3gf.5 . . . . 5 (𝑦 = 𝐵 → (𝜓𝜒))
1413imbi2d 228 . . . 4 (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
15 vtocl3gf.6 . . . . 5 (𝑧 = 𝐶 → (𝜒𝜃))
1615imbi2d 228 . . . 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 2677 . . . 4 (𝐴 ∈ V → 𝜓)
222, 3, 4, 8, 12, 14, 16, 21vtocl2gf 2681 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐴 ∈ V → 𝜃))
231, 22mpan9 275 . 2 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → 𝜃)
24233impb 1139 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  w3a 924   = wceq 1289  wnf 1394  wcel 1438  wnfc 2215  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621
This theorem is referenced by:  vtocl3gaf  2688
  Copyright terms: Public domain W3C validator