Theorem spc2gv 2821

Description: Specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 27-Apr-2004.)

Hypothesis

Ref	Expression
spc2egv.1	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
spc2gv	⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦𝜑 → 𝜓))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑊(𝑥,𝑦)

Proof of Theorem spc2gv

Step	Hyp	Ref	Expression
1		elisset 2744	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		elisset 2744	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
3	1, 2	anim12i 336	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4		eeanv 1925	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
5	3, 4	sylibr 133	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6		spc2egv.1	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
7	6	biimpcd 158	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓))
8	7	2alimi 1449	. . . 4 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓))
9		exim 1592	. . . . 5 ⊢ (∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓) → (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑦𝜓))
10	9	alimi 1448	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓) → ∀𝑥(∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑦𝜓))
11		exim 1592	. . . 4 ⊢ (∀𝑥(∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑦𝜓) → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜓))
12	8, 10, 11	3syl 17	. . 3 ⊢ (∀𝑥∀𝑦𝜑 → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑥∃𝑦𝜓))
13		19.9v 1864	. . . 4 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓)
14		19.9v 1864	. . . 4 ⊢ (∃𝑦𝜓 ↔ 𝜓)
15	13, 14	bitri 183	. . 3 ⊢ (∃𝑥∃𝑦𝜓 ↔ 𝜓)
16	12, 15	syl6ib 160	. 2 ⊢ (∀𝑥∀𝑦𝜑 → (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓))
17	5, 16	syl5com 29	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦𝜑 → 𝜓))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1346   = wceq 1348  ∃wex 1485   ∈ wcel 2141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732

This theorem is referenced by:  rspc2gv  2846  trel  4094  exmidundif  4192  exmidundifim  4193  elovmpo  6050  cnmpt12  13081  cnmpt22  13088  exmidsbthrlem  14054