Theorem spc2gv 2897

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 2817	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥 𝑥 = 𝐴)
2		elisset 2817	. . . 4 ⊢ (𝐵 ∈ 𝑊 → ∃𝑦 𝑦 = 𝐵)
3	1, 2	anim12i 338	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
4		eeanv 1985	. . 3 ⊢ (∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) ↔ (∃𝑥 𝑥 = 𝐴 ∧ ∃𝑦 𝑦 = 𝐵))
5	3, 4	sylibr 134	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → ∃𝑥∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵))
6		spc2egv.1	. . . . . 6 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
7	6	biimpcd 159	. . . . 5 ⊢ (𝜑 → ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓))
8	7	2alimi 1504	. . . 4 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓))
9		exim 1647	. . . . 5 ⊢ (∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓) → (∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑦𝜓))
10	9	alimi 1503	. . . 4 ⊢ (∀𝑥∀𝑦((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → 𝜓) → ∀𝑥(∃𝑦(𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → ∃𝑦𝜓))
11		exim 1647	. . . 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 1919	. . . 4 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓)
14		19.9v 1919	. . . 4 ⊢ (∃𝑦𝜓 ↔ 𝜓)
15	13, 14	bitri 184	. . 3 ⊢ (∃𝑥∃𝑦𝜓 ↔ 𝜓)
16	12, 15	imbitrdi 161	. 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 104   ↔ wb 105  ∀wal 1395   = wceq 1397  ∃wex 1540   ∈ wcel 2202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-v 2804

This theorem is referenced by:  rspc2gv  2922  trel  4194  exmidundif  4296  exmidundifim  4297  elovmpo  6220  seqf1oglem2  10781  seqf1og  10782  cnmpt12  15010  cnmpt22  15017  exmidsbthrlem  16626