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Theorem spc3egv 3553
 Description: Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) Avoid ax-10 2142 and ax-11 2158. (Revised by Gino Giotto, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Aug-2023.)
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 elex 3460 . 2 (𝐴𝑉𝐴 ∈ V)
2 elex 3460 . 2 (𝐵𝑊𝐵 ∈ V)
3 elex 3460 . 2 (𝐶𝑋𝐶 ∈ V)
4 simp1 1133 . . 3 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐴 ∈ V)
5 spc3egv.1 . . . . . . . . . . 11 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
653coml 1124 . . . . . . . . . 10 ((𝑦 = 𝐵𝑧 = 𝐶𝑥 = 𝐴) → (𝜑𝜓))
763expa 1115 . . . . . . . . 9 (((𝑦 = 𝐵𝑧 = 𝐶) ∧ 𝑥 = 𝐴) → (𝜑𝜓))
87pm5.74da 803 . . . . . . . 8 ((𝑦 = 𝐵𝑧 = 𝐶) → ((𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴𝜓)))
98spc2egv 3549 . . . . . . 7 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝑥 = 𝐴𝜓) → ∃𝑦𝑧(𝑥 = 𝐴𝜑)))
10 19.37v 1998 . . . . . . . . 9 (∃𝑧(𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴 → ∃𝑧𝜑))
1110exbii 1849 . . . . . . . 8 (∃𝑦𝑧(𝑥 = 𝐴𝜑) ↔ ∃𝑦(𝑥 = 𝐴 → ∃𝑧𝜑))
12 19.37v 1998 . . . . . . . 8 (∃𝑦(𝑥 = 𝐴 → ∃𝑧𝜑) ↔ (𝑥 = 𝐴 → ∃𝑦𝑧𝜑))
1311, 12bitri 278 . . . . . . 7 (∃𝑦𝑧(𝑥 = 𝐴𝜑) ↔ (𝑥 = 𝐴 → ∃𝑦𝑧𝜑))
149, 13syl6ib 254 . . . . . 6 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝑥 = 𝐴𝜓) → (𝑥 = 𝐴 → ∃𝑦𝑧𝜑)))
1514pm2.86d 108 . . . . 5 ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝑥 = 𝐴 → (𝜓 → ∃𝑦𝑧𝜑)))
16153adant1 1127 . . . 4 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝑥 = 𝐴 → (𝜓 → ∃𝑦𝑧𝜑)))
1716imp 410 . . 3 (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑥 = 𝐴) → (𝜓 → ∃𝑦𝑧𝜑))
184, 17spcimedv 3543 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝜓 → ∃𝑥𝑦𝑧𝜑))
191, 2, 3, 18syl3an 1157 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (𝜓 → ∃𝑥𝑦𝑧𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111  Vcvv 3442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-3an 1086  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444 This theorem is referenced by:  spc3gv  3554  dihjatcclem4  38868  fundcmpsurbijinjpreimafv  44092  fundcmpsurinjALT  44097
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