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Theorem vtocl2 2715
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 2668 . . . . 5 𝑥 𝑥 = 𝐴
3 vtocl2.2 . . . . . 6 𝐵 ∈ V
43isseti 2668 . . . . 5 𝑦 𝑦 = 𝐵
5 eeanv 1884 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6 vtocl2.3 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
76biimpd 143 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
872eximi 1565 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
95, 8sylbir 134 . . . . 5 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
102, 4, 9mp2an 422 . . . 4 𝑥𝑦(𝜑𝜓)
11 nfv 1493 . . . . 5 𝑦𝜓
121119.36-1 1636 . . . 4 (∃𝑦(𝜑𝜓) → (∀𝑦𝜑𝜓))
1310, 12eximii 1566 . . 3 𝑥(∀𝑦𝜑𝜓)
141319.36aiv 1857 . 2 (∀𝑥𝑦𝜑𝜓)
15 vtocl2.4 . . 3 𝜑
1615ax-gen 1410 . 2 𝑦𝜑
1714, 16mpg 1412 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1314   = wceq 1316  wex 1453  wcel 1465  Vcvv 2660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-v 2662
This theorem is referenced by:  undifexmid  4087  caovord  5910  ctssexmid  6992
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