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Theorem spc3gv 2711
Description: Specialization with 3 quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.)
Hypothesis
Ref Expression
spc3egv.1 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
Assertion
Ref Expression
spc3gv ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥𝑦𝑧𝜑𝜓))
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)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem spc3gv
StepHypRef Expression
1 elisset 2633 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2633 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 elisset 2633 . . . 4 (𝐶𝑋 → ∃𝑧 𝑧 = 𝐶)
41, 2, 33anim123i 1128 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
5 eeeanv 1856 . . 3 (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
64, 5sylibr 132 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶))
7 spc3egv.1 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
87biimpcd 157 . . . . . . 7 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓))
982alimi 1390 . . . . . 6 (∀𝑦𝑧𝜑 → ∀𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓))
109alimi 1389 . . . . 5 (∀𝑥𝑦𝑧𝜑 → ∀𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓))
11 exim 1535 . . . . . 6 (∀𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓) → (∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜓))
12112alimi 1390 . . . . 5 (∀𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓) → ∀𝑥𝑦(∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜓))
1310, 12syl 14 . . . 4 (∀𝑥𝑦𝑧𝜑 → ∀𝑥𝑦(∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜓))
14 exim 1535 . . . . 5 (∀𝑦(∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜓) → (∃𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑦𝑧𝜓))
1514alimi 1389 . . . 4 (∀𝑥𝑦(∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜓) → ∀𝑥(∃𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑦𝑧𝜓))
16 exim 1535 . . . 4 (∀𝑥(∃𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑦𝑧𝜓) → (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑥𝑦𝑧𝜓))
1713, 15, 163syl 17 . . 3 (∀𝑥𝑦𝑧𝜑 → (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑥𝑦𝑧𝜓))
18 19.9v 1799 . . . 4 (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑦𝑧𝜓)
19 19.9v 1799 . . . 4 (∃𝑦𝑧𝜓 ↔ ∃𝑧𝜓)
20 19.9v 1799 . . . 4 (∃𝑧𝜓𝜓)
2118, 19, 203bitri 204 . . 3 (∃𝑥𝑦𝑧𝜓𝜓)
2217, 21syl6ib 159 . 2 (∀𝑥𝑦𝑧𝜑 → (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓))
236, 22syl5com 29 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∀𝑥𝑦𝑧𝜑𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  w3a 924  wal 1287   = wceq 1289  wex 1426  wcel 1438
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  funopg  5042
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