Theorem spc2gv 3555

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 3554	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → (¬ 𝜓 → ∃𝑥∃𝑦 ¬ 𝜑))
4		2nalexn 1861	. . 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))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-clel 2836

This theorem is used by:  rspc2gv  3586  trel  5220  elovmpo  7666  seqf1olem2  14185  seqf1o  14186  fi1uzind  14652  brfi1indALT  14655  pslem  18746  cnmpt12  23986  cnmpt22  23993  mclsppslem  36348  mbfresfi  38584  lpolconN  42544  ismrcd2  43709  ismrc  43711  euendfunc  50633