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Theorem vtocl3 3411
 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 3358 . . . . . 6 𝑥 𝑥 = 𝐴
3 vtocl3.2 . . . . . . 7 𝐵 ∈ V
43isseti 3358 . . . . . 6 𝑦 𝑦 = 𝐵
5 vtocl3.3 . . . . . . 7 𝐶 ∈ V
65isseti 3358 . . . . . 6 𝑧 𝑧 = 𝐶
7 eeeanv 2344 . . . . . . 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 219 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
109eximi 1909 . . . . . . . 8 (∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧(𝜑𝜓))
11102eximi 1910 . . . . . . 7 (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑥𝑦𝑧(𝜑𝜓))
127, 11sylbir 225 . . . . . 6 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶) → ∃𝑥𝑦𝑧(𝜑𝜓))
132, 4, 6, 12mp3an 1571 . . . . 5 𝑥𝑦𝑧(𝜑𝜓)
14 19.36v 2071 . . . . . 6 (∃𝑧(𝜑𝜓) ↔ (∀𝑧𝜑𝜓))
15142exbii 1924 . . . . 5 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥𝑦(∀𝑧𝜑𝜓))
1613, 15mpbi 220 . . . 4 𝑥𝑦(∀𝑧𝜑𝜓)
17 19.36v 2071 . . . . 5 (∃𝑦(∀𝑧𝜑𝜓) ↔ (∀𝑦𝑧𝜑𝜓))
1817exbii 1923 . . . 4 (∃𝑥𝑦(∀𝑧𝜑𝜓) ↔ ∃𝑥(∀𝑦𝑧𝜑𝜓))
1916, 18mpbi 220 . . 3 𝑥(∀𝑦𝑧𝜑𝜓)
201919.36iv 2025 . 2 (∀𝑥𝑦𝑧𝜑𝜓)
21 vtocl3.5 . . 3 𝜑
2221gen2 1870 . 2 𝑦𝑧𝜑
2320, 22mpg 1871 1 𝜓
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ w3a 1070  ∀wal 1628   = wceq 1630  ∃wex 1851   ∈ wcel 2144  Vcvv 3349 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-v 3351 This theorem is referenced by: (None)
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