Theorem cla42gv 1399

Description: Specialization with 2 quantifiers, using implicit substitution.

Hypothesis

Ref	Expression
cla4e2gv.1	⊢ ((x = A ∧ y = B) → (φ ↔ ψ))

Assertion

Ref	Expression
cla42gv	⊢ ((A ∈ C ∧ B ∈ D) → (∀x∀yφ → ψ))

Distinct variable group(s):   x,y,A   x,B,y   ψ,x,y

Proof of Theorem cla42gv

Step	Hyp	Ref	Expression
1		cla4e2gv.1	. . . . 5 ⊢ ((x = A ∧ y = B) → (φ ↔ ψ))
2	1	negbid 463	. . . 4 ⊢ ((x = A ∧ y = B) → (¬ φ ↔ ¬ ψ))
3	2	cla4e2gv 1398	. . 3 ⊢ ((A ∈ C ∧ B ∈ D) → (¬ ψ → ∃x∃y ¬ φ))
4		exnal 721	. . . . 5 ⊢ (∃y ¬ φ ↔ ¬ ∀yφ)
5	4	biex 733	. . . 4 ⊢ (∃x∃y ¬ φ ↔ ∃x ¬ ∀yφ)
6		exnal 721	. . . 4 ⊢ (∃x ¬ ∀yφ ↔ ¬ ∀x∀yφ)
7	5, 6	bitr2 152	. . 3 ⊢ (¬ ∀x∀yφ ↔ ∃x∃y ¬ φ)
8	3, 7	syl6ibr 186	. 2 ⊢ ((A ∈ C ∧ B ∈ D) → (¬ ψ → ¬ ∀x∀yφ))
9	8	a3d 70	1 ⊢ ((A ∈ C ∧ B ∈ D) → (∀x∀yφ → ψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-9 799  ax-12 802  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349

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