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Theorem vtocl3 2808
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 2760 . . . . . 6 𝑥 𝑥 = 𝐴
3 vtocl3.2 . . . . . . 7 𝐵 ∈ V
43isseti 2760 . . . . . 6 𝑦 𝑦 = 𝐵
5 vtocl3.3 . . . . . . 7 𝐶 ∈ V
65isseti 2760 . . . . . 6 𝑧 𝑧 = 𝐶
7 eeeanv 1945 . . . . . . 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 144 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
109eximi 1611 . . . . . . . 8 (∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧(𝜑𝜓))
11102eximi 1612 . . . . . . 7 (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑥𝑦𝑧(𝜑𝜓))
127, 11sylbir 135 . . . . . 6 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶) → ∃𝑥𝑦𝑧(𝜑𝜓))
132, 4, 6, 12mp3an 1348 . . . . 5 𝑥𝑦𝑧(𝜑𝜓)
14 nfv 1539 . . . . . . 7 𝑧𝜓
151419.36-1 1684 . . . . . 6 (∃𝑧(𝜑𝜓) → (∀𝑧𝜑𝜓))
16152eximi 1612 . . . . 5 (∃𝑥𝑦𝑧(𝜑𝜓) → ∃𝑥𝑦(∀𝑧𝜑𝜓))
1713, 16ax-mp 5 . . . 4 𝑥𝑦(∀𝑧𝜑𝜓)
18 nfv 1539 . . . . 5 𝑦𝜓
191819.36-1 1684 . . . 4 (∃𝑦(∀𝑧𝜑𝜓) → (∀𝑦𝑧𝜑𝜓))
2017, 19eximii 1613 . . 3 𝑥(∀𝑦𝑧𝜑𝜓)
212019.36aiv 1913 . 2 (∀𝑥𝑦𝑧𝜑𝜓)
22 vtocl3.5 . . 3 𝜑
2322gen2 1461 . 2 𝑦𝑧𝜑
2421, 23mpg 1462 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105  w3a 980  wal 1362   = wceq 1364  wex 1503  wcel 2160  Vcvv 2752
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-v 2754
This theorem is referenced by: (None)
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