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Theorem vtocl2 2626
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 2580 . . . . 5 𝑥 𝑥 = 𝐴
3 vtocl2.2 . . . . . 6 𝐵 ∈ V
43isseti 2580 . . . . 5 𝑦 𝑦 = 𝐵
5 eeanv 1823 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
6 vtocl2.3 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
76biimpd 136 . . . . . . 7 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
872eximi 1508 . . . . . 6 (∃𝑥𝑦(𝑥 = 𝐴𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
95, 8sylbir 129 . . . . 5 ((∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵) → ∃𝑥𝑦(𝜑𝜓))
102, 4, 9mp2an 410 . . . 4 𝑥𝑦(𝜑𝜓)
11 nfv 1437 . . . . 5 𝑦𝜓
121119.36-1 1579 . . . 4 (∃𝑦(𝜑𝜓) → (∀𝑦𝜑𝜓))
1310, 12eximii 1509 . . 3 𝑥(∀𝑦𝜑𝜓)
141319.36aiv 1797 . 2 (∀𝑥𝑦𝜑𝜓)
15 vtocl2.4 . . 3 𝜑
1615ax-gen 1354 . 2 𝑦𝜑
1714, 16mpg 1356 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102  wal 1257   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  caovord  5700
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