Theorem spc2egv 2636

Description: Existential specialization with 2 quantifiers, using implicit substitution. (Contributed by NM, 3-Aug-1995.)

Hypothesis

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

Assertion

Ref	Expression
spc2egv	⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (ψ → ∃x∃yφ))

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 spc2egv

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	. . . 4 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
7	6	biimprcd 149	. . 3 ⊢ (ψ → ((x = A ∧ y = B) → φ))
8	7	2eximdv 1759	. 2 ⊢ (ψ → (∃x∃y(x = A ∧ y = B) → ∃x∃yφ))
9	5, 8	syl5com 26	1 ⊢ ((A ∈ 𝑉 ∧ B ∈ 𝑊) → (ψ → ∃x∃yφ))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = 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:  spc2ev  2642  th3q  6147  addnnnq0  6432  mulnnnq0  6433  addsrpr  6673  mulsrpr  6674