Theorem spc3egv 3542

Description: Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) Avoid ax-10 2137 and ax-11 2154. (Revised by Gino Giotto, 20-Aug-2023.) (Proof shortened by Wolf Lammen, 25-Aug-2023.)

Hypothesis

Ref	Expression
spc3egv.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spc3egv	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑))

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

Step	Hyp	Ref	Expression
1		elex 3450	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3450	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		elex 3450	. 2 ⊢ (𝐶 ∈ 𝑋 → 𝐶 ∈ V)
4		simp1 1135	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐴 ∈ V)
5		spc3egv.1	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
6	5	3coml 1126	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
7	6	3expa 1117	. . . . . . . . 9 ⊢ (((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
8	7	pm5.74da 801	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)))
9	8	spc2egv 3538	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝑥 = 𝐴 → 𝜓) → ∃𝑦∃𝑧(𝑥 = 𝐴 → 𝜑)))
10		19.37v 1995	. . . . . . . . 9 ⊢ (∃𝑧(𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → ∃𝑧𝜑))
11	10	exbii 1850	. . . . . . . 8 ⊢ (∃𝑦∃𝑧(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 → ∃𝑧𝜑))
12		19.37v 1995	. . . . . . . 8 ⊢ (∃𝑦(𝑥 = 𝐴 → ∃𝑧𝜑) ↔ (𝑥 = 𝐴 → ∃𝑦∃𝑧𝜑))
13	11, 12	bitri 274	. . . . . . 7 ⊢ (∃𝑦∃𝑧(𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → ∃𝑦∃𝑧𝜑))
14	9, 13	syl6ib 250	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝑥 = 𝐴 → 𝜓) → (𝑥 = 𝐴 → ∃𝑦∃𝑧𝜑)))
15	14	pm2.86d 108	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝑥 = 𝐴 → (𝜓 → ∃𝑦∃𝑧𝜑)))
16	15	3adant1 1129	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝑥 = 𝐴 → (𝜓 → ∃𝑦∃𝑧𝜑)))
17	16	imp 407	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑥 = 𝐴) → (𝜓 → ∃𝑦∃𝑧𝜑))
18	4, 17	spcimedv 3534	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑))
19	1, 2, 3, 18	syl3an 1159	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-3an 1088  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434

This theorem is referenced by:  spc3gv  3543  dihjatcclem4  39435  fundcmpsurbijinjpreimafv  44859  fundcmpsurinjALT  44864