Theorem spc2gv 2943

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		spc2egv.1	. . . . 5 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
2	1	notbid 285	. . . 4 ⊢ ((x = A ∧ y = B) → (¬ φ ↔ ¬ ψ))
3	2	spc2egv 2942	. . 3 ⊢ ((A ∈ V ∧ B ∈ W) → (¬ ψ → ∃x∃y ¬ φ))
4		2nalexn 1573	. . 3 ⊢ (¬ ∀x∀yφ ↔ ∃x∃y ¬ φ)
5	3, 4	syl6ibr 218	. 2 ⊢ ((A ∈ V ∧ B ∈ W) → (¬ ψ → ¬ ∀x∀yφ))
6	5	con4d 97	1 ⊢ ((A ∈ V ∧ B ∈ W) → (∀x∀yφ → ψ))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2862

This theorem is referenced by:  nnsucelr  4429  ssfin  4471  ncfinlower  4484