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Theorem vtocl3gf 2633
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 2583 . . 3 (𝐴𝑉𝐴 ∈ V)
2 vtocl3gf.d . . . 4 𝑦𝐵
3 vtocl3gf.e . . . 4 𝑧𝐵
4 vtocl3gf.f . . . 4 𝑧𝐶
5 vtocl3gf.b . . . . . 6 𝑦𝐴
65nfel1 2204 . . . . 5 𝑦 𝐴 ∈ V
7 vtocl3gf.2 . . . . 5 𝑦𝜒
86, 7nfim 1480 . . . 4 𝑦(𝐴 ∈ V → 𝜒)
9 vtocl3gf.c . . . . . 6 𝑧𝐴
109nfel1 2204 . . . . 5 𝑧 𝐴 ∈ V
11 vtocl3gf.3 . . . . 5 𝑧𝜃
1210, 11nfim 1480 . . . 4 𝑧(𝐴 ∈ V → 𝜃)
13 vtocl3gf.5 . . . . 5 (𝑦 = 𝐵 → (𝜓𝜒))
1413imbi2d 223 . . . 4 (𝑦 = 𝐵 → ((𝐴 ∈ V → 𝜓) ↔ (𝐴 ∈ V → 𝜒)))
15 vtocl3gf.6 . . . . 5 (𝑧 = 𝐶 → (𝜒𝜃))
1615imbi2d 223 . . . 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 2629 . . . 4 (𝐴 ∈ V → 𝜓)
222, 3, 4, 8, 12, 14, 16, 21vtocl2gf 2632 . . 3 ((𝐵𝑊𝐶𝑋) → (𝐴 ∈ V → 𝜃))
231, 22mpan9 269 . 2 ((𝐴𝑉 ∧ (𝐵𝑊𝐶𝑋)) → 𝜃)
24233impb 1111 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → 𝜃)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  w3a 896   = wceq 1259  wnf 1365  wcel 1409  wnfc 2181  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576
This theorem is referenced by:  vtocl3gaf  2639
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