Theorem spc2d 3541

Description: 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
spc2d	⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∀𝑥∀𝑦𝜓 → 𝜒))

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 spc2d

Step	Hyp	Ref	Expression
1		2nalexn 1830	. . 3 ⊢ (¬ ∀𝑥∀𝑦𝜓 ↔ ∃𝑥∃𝑦 ¬ 𝜓)
2	1	con1bii 357	. 2 ⊢ (¬ ∃𝑥∃𝑦 ¬ 𝜓 ↔ ∀𝑥∀𝑦𝜓)
3		spc2ed.x	. . . . 5 ⊢ Ⅎ𝑥𝜒
4	3	nfn 1860	. . . 4 ⊢ Ⅎ𝑥 ¬ 𝜒
5		spc2ed.y	. . . . 5 ⊢ Ⅎ𝑦𝜒
6	5	nfn 1860	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝜒
7		spc2ed.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
8	7	notbid 318	. . . 4 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (¬ 𝜓 ↔ ¬ 𝜒))
9	4, 6, 8	spc2ed 3540	. . 3 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (¬ 𝜒 → ∃𝑥∃𝑦 ¬ 𝜓))
10	9	con1d 145	. 2 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (¬ ∃𝑥∃𝑦 ¬ 𝜓 → 𝜒))
11	2, 10	syl5bir 242	1 ⊢ ((𝜑 ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊)) → (∀𝑥∀𝑦𝜓 → 𝜒))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782  Ⅎwnf 1786   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-clel 2816

This theorem is referenced by: (None)