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Theorem spc3egv 3499
Description: Existential specialization with three 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 3417 . . . 4 (𝐴𝑉 → ∃𝑥 𝑥 = 𝐴)
2 elisset 3417 . . . 4 (𝐵𝑊 → ∃𝑦 𝑦 = 𝐵)
3 elisset 3417 . . . 4 (𝐶𝑋 → ∃𝑧 𝑧 = 𝐶)
41, 2, 33anim123i 1151 . . 3 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
5 eeeanv 2319 . . 3 (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵 ∧ ∃𝑧 𝑧 = 𝐶))
64, 5sylibr 226 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → ∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶))
7 spc3egv.1 . . . . 5 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
87biimprcd 242 . . . 4 (𝜓 → ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → 𝜑))
98eximdv 1960 . . 3 (𝜓 → (∃𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑧𝜑))
1092eximdv 1962 . 2 (𝜓 → (∃𝑥𝑦𝑧(𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → ∃𝑥𝑦𝑧𝜑))
116, 10syl5com 31 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → ∃𝑥𝑦𝑧𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1071   = wceq 1601  wex 1823  wcel 2107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-v 3400
This theorem is referenced by:  spc3gv  3500  dihjatcclem4  37580
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