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Theorem vtocl2 3247
Description: Implicit substitution of classes for setvar variables. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 8-Jun-2011.)
Hypotheses
Ref Expression
vtocl2.1 𝐴 ∈ V
vtocl2.2 𝐵 ∈ V
vtocl2.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
vtocl2.4 𝜑
Assertion
Ref Expression
vtocl2 𝜓
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem vtocl2
StepHypRef Expression
1 vtocl2.1 . . . . . 6 𝐴 ∈ V
21isseti 3195 . . . . 5 𝑥 𝑥 = 𝐴
3 vtocl2.2 . . . . . 6 𝐵 ∈ V
43isseti 3195 . . . . 5 𝑦 𝑦 = 𝐵
5 eeanv 2181 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6 vtocl2.3 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
76biimpd 219 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
872eximi 1760 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
95, 8sylbir 225 . . . . 5 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
102, 4, 9mp2an 707 . . . 4 𝑥𝑦(𝜑𝜓)
11 19.36v 1901 . . . . 5 (∃𝑦(𝜑𝜓) ↔ (∀𝑦𝜑𝜓))
1211exbii 1771 . . . 4 (∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(∀𝑦𝜑𝜓))
1310, 12mpbi 220 . . 3 𝑥(∀𝑦𝜑𝜓)
141319.36iv 1902 . 2 (∀𝑥𝑦𝜑𝜓)
15 vtocl2.4 . . 3 𝜑
1615ax-gen 1719 . 2 𝑦𝜑
1714, 16mpg 1721 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  wal 1478   = wceq 1480  wex 1701  wcel 1987  Vcvv 3186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-v 3188
This theorem is referenced by:  caovord  6798  sornom  9043  wloglei  10504  ipodrsima  17086  mpfind  19455  mclsppslem  31188  monotoddzzfi  36987
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