Theorem spc2ed 3569

Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by Thierry Arnoux, 23-Aug-2017.)

Hypotheses

Ref	Expression
spc2ed.x	⊢ Ⅎ𝑥𝜒
spc2ed.y	⊢ Ⅎ𝑦𝜒
spc2ed.1	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
spc2ed	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝜒 → ∃𝑥∃𝑦𝜓))

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem spc2ed

Step	Hyp	Ref	Expression
1		elisset 2851	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		elisset 2851	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
3	1, 2	anim12i 624	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4		exdistrv 1982	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
5	3, 4	sylibr 237	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6		nfv 1941	. . . . 5 ⊢ Ⅎ𝑥𝜑
7		spc2ed.x	. . . . 5 ⊢ Ⅎ𝑥𝜒
8	6, 7	nfan 1926	. . . 4 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜒)
9		nfv 1941	. . . . . 6 ⊢ Ⅎ𝑦𝜑
10		spc2ed.y	. . . . . 6 ⊢ Ⅎ𝑦𝜒
11	9, 10	nfan 1926	. . . . 5 ⊢ Ⅎ𝑦(𝜑 ∧ 𝜒)
12		anass 473	. . . . . . . 8 ⊢ (((𝜒 ∧ 𝜑) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ↔ (𝜒 ∧ (𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))))
13		ancom 465	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝜑) ↔ (𝜑 ∧ 𝜒))
14	13	anbi1i 635	. . . . . . . 8 ⊢ (((𝜒 ∧ 𝜑) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) ↔ ((𝜑 ∧ 𝜒) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
15	12, 14	bitr3i 280	. . . . . . 7 ⊢ ((𝜒 ∧ (𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))) ↔ ((𝜑 ∧ 𝜒) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)))
16		spc2ed.1	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
17	16	biimparc 484	. . . . . . 7 ⊢ ((𝜒 ∧ (𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵))) → 𝜓)
18	15, 17	sylbir 238	. . . . . 6 ⊢ (((𝜑 ∧ 𝜒) ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → 𝜓)
19	18	ex 417	. . . . 5 ⊢ ((𝜑 ∧ 𝜒) → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓))
20	11, 19	eximd 2258	. . . 4 ⊢ ((𝜑 ∧ 𝜒) → (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑦𝜓))
21	8, 20	eximd 2258	. . 3 ⊢ ((𝜑 ∧ 𝜒) → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜓))
22	21	impancom 456	. 2 ⊢ ((𝜑 ∧ ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜒 → ∃𝑥∃𝑦𝜓))
23	5, 22	sylan2 604	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (𝜒 → ∃𝑥∃𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1567  ∃wex 1806  Ⅎwnf 1810   ∈ wcel 2149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-12 2219

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-clel 2844

This theorem is referenced by:  spc2d  3570  or2expropbilem1  47651  ich2exprop  48102  reuopreuprim  48157