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Theorem spc3egv 2690
Description: Existential 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
spc3egv ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → ∃𝑥𝑦𝑧𝜑))
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 spc3egv
StepHypRef Expression
1 elisset 2614 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2614 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 elisset 2614 . . . 4 (𝐶𝑋 → ∃𝑧 𝑧 = 𝐶)
41, 2, 33anim123i 1124 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
5 eeeanv 1850 . . 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 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
87biimprcd 158 . . . 4 (𝜓 → ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜑))
98eximdv 1802 . . 3 (𝜓 → (∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜑))
1092eximdv 1804 . 2 (𝜓 → (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑥𝑦𝑧𝜑))
116, 10syl5com 29 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → ∃𝑥𝑦𝑧𝜑))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  w3a 920   = wceq 1285  wex 1422  wcel 1434
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 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-3an 922  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-v 2604
This theorem is referenced by: (None)
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