Theorem spc2gv 2637

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 2562	. . . 4 ⊢ (A ∈ 𝑉 → ∃x x = A)
2		elisset 2562	. . . 4 ⊢ (B ∈ 𝑊 → ∃y y = B)
3	1, 2	anim12i 321	. . 3 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (∃x x = A ∧ ∃y y = B))
4		eeanv 1804	. . 3 ⊢ (∃x∃y(x = A ∧ y = B) ↔ (∃x x = A ∧ ∃y y = B))
5	3, 4	sylibr 137	. 2 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → ∃x∃y(x = A ∧ y = B))
6		spc2egv.1	. . . . . 6 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
7	6	biimpcd 148	. . . . 5 ⊢ (φ → ((x = A ∧ y = B) → ψ))
8	7	2alimi 1342	. . . 4 ⊢ (∀x∀yφ → ∀x∀y((x = A ∧ y = B) → ψ))
9		exim 1487	. . . . 5 ⊢ (∀y((x = A ∧ y = B) → ψ) → (∃y(x = A ∧ y = B) → ∃yψ))
10	9	alimi 1341	. . . 4 ⊢ (∀x∀y((x = A ∧ y = B) → ψ) → ∀x(∃y(x = A ∧ y = B) → ∃yψ))
11		exim 1487	. . . 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 1748	. . . 4 ⊢ (∃x∃yψ ↔ ∃yψ)
14		19.9v 1748	. . . 4 ⊢ (∃yψ ↔ ψ)
15	13, 14	bitri 173	. . 3 ⊢ (∃x∃yψ ↔ ψ)
16	12, 15	syl6ib 150	. 2 ⊢ (∀x∀yφ → (∃x∃y(x = A ∧ y = B) → ψ))
17	5, 16	syl5com 26	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (∀x∀yφ → ψ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   = wceq 1242  ∃wex 1378   ∈ wcel 1390

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553

This theorem is referenced by:  trel  3852  elovmpt2  5643