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Theorem vtocl3 3478
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 3426 . . . . . 6 𝑥 𝑥 = 𝐴
3 vtocl3.2 . . . . . . 7 𝐵 ∈ V
43isseti 3426 . . . . . 6 𝑦 𝑦 = 𝐵
5 vtocl3.3 . . . . . . 7 𝐶 ∈ V
65isseti 3426 . . . . . 6 𝑧 𝑧 = 𝐶
7 eeeanv 2374 . . . . . . 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 221 . . . . . . . . 9 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
109eximi 1933 . . . . . . . 8 (∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧(𝜑𝜓))
11102eximi 1934 . . . . . . 7 (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑥𝑦𝑧(𝜑𝜓))
127, 11sylbir 227 . . . . . 6 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶) → ∃𝑥𝑦𝑧(𝜑𝜓))
132, 4, 6, 12mp3an 1589 . . . . 5 𝑥𝑦𝑧(𝜑𝜓)
14 19.36v 2091 . . . . . 6 (∃𝑧(𝜑𝜓) ↔ (∀𝑧𝜑𝜓))
15142exbii 1948 . . . . 5 (∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥𝑦(∀𝑧𝜑𝜓))
1613, 15mpbi 222 . . . 4 𝑥𝑦(∀𝑧𝜑𝜓)
17 19.36v 2091 . . . . 5 (∃𝑦(∀𝑧𝜑𝜓) ↔ (∀𝑦𝑧𝜑𝜓))
1817exbii 1947 . . . 4 (∃𝑥𝑦(∀𝑧𝜑𝜓) ↔ ∃𝑥(∀𝑦𝑧𝜑𝜓))
1916, 18mpbi 222 . . 3 𝑥(∀𝑦𝑧𝜑𝜓)
201919.36iv 2045 . 2 (∀𝑥𝑦𝑧𝜑𝜓)
21 vtocl3.5 . . 3 𝜑
2221gen2 1895 . 2 𝑦𝑧𝜑
2320, 22mpg 1896 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1111  wal 1654   = wceq 1656  wex 1878  wcel 2164  Vcvv 3414
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-ext 2803
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-clab 2812  df-cleq 2818  df-clel 2821  df-v 3416
This theorem is referenced by: (None)
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