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Theorem spc3gv 2662
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 2585 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2585 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 elisset 2585 . . . 4 (𝐶𝑋 → ∃𝑧 𝑧 = 𝐶)
41, 2, 33anim123i 1100 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
5 eeeanv 1824 . . 3 (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
64, 5sylibr 141 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶))
7 spc3egv.1 . . . . . . . 8 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
87biimpcd 152 . . . . . . 7 (𝜑 → ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓))
982alimi 1361 . . . . . 6 (∀𝑦𝑧𝜑 → ∀𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓))
109alimi 1360 . . . . 5 (∀𝑥𝑦𝑧𝜑 → ∀𝑥𝑦𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓))
11 exim 1506 . . . . . 6 (∀𝑧((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜓) → (∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜓))
12112alimi 1361 . . . . 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 1506 . . . . 5 (∀𝑦(∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜓) → (∃𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑦𝑧𝜓))
1514alimi 1360 . . . 4 (∀𝑥𝑦(∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜓) → ∀𝑥(∃𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑦𝑧𝜓))
16 exim 1506 . . . 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 1767 . . . 4 (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑦𝑧𝜓)
19 19.9v 1767 . . . 4 (∃𝑦𝑧𝜓 ↔ ∃𝑧𝜓)
20 19.9v 1767 . . . 4 (∃𝑧𝜓𝜓)
2118, 19, 203bitri 199 . . 3 (∃𝑥𝑦𝑧𝜓𝜓)
2217, 21syl6ib 154 . 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 102  w3a 896  wal 1257   = wceq 1259  wex 1397  wcel 1409
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-3an 898  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  funopg  4962
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