Theorem spc2gv 3568

Description: Specialization with two 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		spc2egv.1	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
2	1	notbid 321	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (¬ 𝜑 ↔ ¬ 𝜓))
3	2	spc2egv 3567	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝜓 → ∃𝑥∃𝑦 ¬ 𝜑))
4		2nalexn 1855	. . 3 ⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑)
5	3, 4	imbitrrdi 255	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝜓 → ¬ ∀𝑥∀𝑦𝜑))
6	5	con4d 116	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (∀𝑥∀𝑦𝜑 → 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565   = wceq 1567  ∃wex 1806   ∈ wcel 2149

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-ex 1807  df-sb 2098  df-clab 2748  df-clel 2844

This theorem is referenced by:  rspc2gv  3600  trel  5230  elovmpo  7656  seqf1olem2  14078  seqf1o  14079  fi1uzind  14544  brfi1indALT  14547  pslem  18628  cnmpt12  23793  cnmpt22  23800  mclsppslem  35974  mbfresfi  38205  lpolconN  42151  ismrcd2  43322  ismrc  43324  euendfunc  50189