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Theorem spc3egv 2772
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 2695 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 2695 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 elisset 2695 . . . 4 (𝐶𝑋 → ∃𝑧 𝑧 = 𝐶)
41, 2, 33anim123i 1166 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
5 eeeanv 1903 . . 3 (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
64, 5sylibr 133 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶))
7 spc3egv.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
87biimprcd 159 . . . 4 (𝜓 → ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜑))
98eximdv 1852 . . 3 (𝜓 → (∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜑))
1092eximdv 1854 . 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 104  w3a 962   = wceq 1331  wex 1468  wcel 1480
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 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-3an 964  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-v 2683
This theorem is referenced by: (None)
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