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Theorem vtocl3 2780
Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 3-Jun-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
vtocl3.1 𝐴 ∈ V
vtocl3.2 𝐵 ∈ V
vtocl3.3 𝐶 ∈ V
vtocl3.4 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
vtocl3.5 𝜑
Assertion
Ref Expression
vtocl3 𝜓
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem vtocl3
StepHypRef Expression
1 vtocl3.1 . . . . . . 7 𝐴 ∈ V
21isseti 2732 . . . . . 6 𝑥 𝑥 = 𝐴
3 vtocl3.2 . . . . . . 7 𝐵 ∈ V
43isseti 2732 . . . . . 6 𝑦 𝑦 = 𝐵
5 vtocl3.3 . . . . . . 7 𝐶 ∈ V
65isseti 2732 . . . . . 6 𝑧 𝑧 = 𝐶
7 eeeanv 1920 . . . . . . 7 (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
8 vtocl3.4 . . . . . . . . . 10 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
98biimpd 143 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
109eximi 1587 . . . . . . . 8 (∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧(𝜑𝜓))
11102eximi 1588 . . . . . . 7 (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑥𝑦𝑧(𝜑𝜓))
127, 11sylbir 134 . . . . . 6 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶) → ∃𝑥𝑦𝑧(𝜑𝜓))
132, 4, 6, 12mp3an 1326 . . . . 5 𝑥𝑦𝑧(𝜑𝜓)
14 nfv 1515 . . . . . . 7 𝑧𝜓
151419.36-1 1660 . . . . . 6 (∃𝑧(𝜑𝜓) → (∀𝑧𝜑𝜓))
16152eximi 1588 . . . . 5 (∃𝑥𝑦𝑧(𝜑𝜓) → ∃𝑥𝑦(∀𝑧𝜑𝜓))
1713, 16ax-mp 5 . . . 4 𝑥𝑦(∀𝑧𝜑𝜓)
18 nfv 1515 . . . . 5 𝑦𝜓
191819.36-1 1660 . . . 4 (∃𝑦(∀𝑧𝜑𝜓) → (∀𝑦𝑧𝜑𝜓))
2017, 19eximii 1589 . . 3 𝑥(∀𝑦𝑧𝜑𝜓)
212019.36aiv 1888 . 2 (∀𝑥𝑦𝑧𝜑𝜓)
22 vtocl3.5 . . 3 𝜑
2322gen2 1437 . 2 𝑦𝑧𝜑
2421, 23mpg 1438 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  w3a 967  wal 1340   = wceq 1342  wex 1479  wcel 2135  Vcvv 2724
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-3an 969  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-v 2726
This theorem is referenced by: (None)
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