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Theorem vtocl2OLD 3568
Description: Obsolete proof of vtocl2 3567 as of 23-Aug-2023. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypotheses
Ref Expression
vtocl2.1 𝐴 ∈ V
vtocl2.2 𝐵 ∈ V
vtocl2.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
vtocl2.4 𝜑
Assertion
Ref Expression
vtocl2OLD 𝜓
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem vtocl2OLD
StepHypRef Expression
1 vtocl2.1 . . . . . 6 𝐴 ∈ V
21isseti 3514 . . . . 5 𝑥 𝑥 = 𝐴
3 vtocl2.2 . . . . . 6 𝐵 ∈ V
43isseti 3514 . . . . 5 𝑦 𝑦 = 𝐵
5 exdistrv 1949 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6 vtocl2.3 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
76biimpd 230 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
872eximi 1829 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
95, 8sylbir 236 . . . . 5 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
102, 4, 9mp2an 688 . . . 4 𝑥𝑦(𝜑𝜓)
11 19.36v 1987 . . . . 5 (∃𝑦(𝜑𝜓) ↔ (∀𝑦𝜑𝜓))
1211exbii 1841 . . . 4 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(∀𝑦𝜑𝜓))
1310, 12mpbi 231 . . 3 𝑥(∀𝑦𝜑𝜓)
141319.36iv 1940 . 2 (∀𝑥𝑦𝜑𝜓)
15 vtocl2.4 . . 3 𝜑
1615ax-gen 1789 . 2 𝑦𝜑
1714, 16mpg 1791 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wex 1773  wcel 2107  Vcvv 3500
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774  df-cleq 2819  df-clel 2898
This theorem is referenced by: (None)
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