Theorem spc3egv 3564

Description: Existential specialization with three quantifiers, using implicit substitution. (Contributed by NM, 12-May-2008.) Avoid ax-10 2179 and ax-11 2195. (Revised by GG, 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 3478	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		elex 3478	. 2 ⊢ (𝐵 ∈ 𝑊 → 𝐵 ∈ V)
3		elex 3478	. 2 ⊢ (𝐶 ∈ 𝑋 → 𝐶 ∈ V)
4		simp1 1154	. . 3 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → 𝐴 ∈ V)
5		spc3egv.1	. . . . . . . . . . 11 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → (𝜑 ↔ 𝜓))
6	5	3coml 1145	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶 ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
7	6	3expa 1136	. . . . . . . . 9 ⊢ (((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) ∧ 𝑥 = 𝐴) → (𝜑 ↔ 𝜓))
8	7	pm5.74da 816	. . . . . . . 8 ⊢ ((𝑦 = 𝐵 ∧ 𝑧 = 𝐶) → ((𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → 𝜓)))
9	8	spc2egv 3560	. . . . . . 7 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝑥 = 𝐴 → 𝜓) → ∃𝑦∃𝑧(𝑥 = 𝐴 → 𝜑)))
10		19.37v 2030	. . . . . . . . 9 ⊢ (∃𝑧(𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → ∃𝑧𝜑))
11	10	exbii 1881	. . . . . . . 8 ⊢ (∃𝑦∃𝑧(𝑥 = 𝐴 → 𝜑) ↔ ∃𝑦(𝑥 = 𝐴 → ∃𝑧𝜑))
12		19.37v 2030	. . . . . . . 8 ⊢ (∃𝑦(𝑥 = 𝐴 → ∃𝑧𝜑) ↔ (𝑥 = 𝐴 → ∃𝑦∃𝑧𝜑))
13	11, 12	bitri 278	. . . . . . 7 ⊢ (∃𝑦∃𝑧(𝑥 = 𝐴 → 𝜑) ↔ (𝑥 = 𝐴 → ∃𝑦∃𝑧𝜑))
14	9, 13	imbitrdi 254	. . . . . 6 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → ((𝑥 = 𝐴 → 𝜓) → (𝑥 = 𝐴 → ∃𝑦∃𝑧𝜑)))
15	14	pm2.86d 109	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝑥 = 𝐴 → (𝜓 → ∃𝑦∃𝑧𝜑)))
16	15	3adant1 1148	. . . 4 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝑥 = 𝐴 → (𝜓 → ∃𝑦∃𝑧𝜑)))
17	16	imp 412	. . 3 ⊢ (((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) ∧ 𝑥 = 𝐴) → (𝜓 → ∃𝑦∃𝑧𝜑))
18	4, 17	spcimedv 3556	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑))
19	1, 2, 3, 18	syl3an 1178	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊 ∧ 𝐶 ∈ 𝑋) → (𝜓 → ∃𝑥∃𝑦∃𝑧𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2146  Vcvv 3457

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-v 3459

This theorem is used by:  spc3gv  3565  dihjatcclem4  42255  fundcmpsurbijinjpreimafv  48216  fundcmpsurinjALT  48221