Theorem aaan 794

Description: Rearrange universal quantifiers.

Hypotheses

Ref	Expression
aaan.1	⊢ (φ → ∀yφ)
aaan.2	⊢ (ψ → ∀xψ)

Assertion

Ref	Expression
aaan	⊢ (∀x∀y(φ ∧ ψ) ↔ (∀xφ ∧ ∀yψ))

Proof of Theorem aaan

Step	Hyp	Ref	Expression
1		aaan.1	. . . 4 ⊢ (φ → ∀yφ)
2	1	19.28 751	. . 3 ⊢ (∀y(φ ∧ ψ) ↔ (φ ∧ ∀yψ))
3	2	bial 695	. 2 ⊢ (∀x∀y(φ ∧ ψ) ↔ ∀x(φ ∧ ∀yψ))
4		aaan.2	. . . 4 ⊢ (ψ → ∀xψ)
5	4	hbal 700	. . 3 ⊢ (∀yψ → ∀x∀yψ)
6	5	19.27 750	. 2 ⊢ (∀x(φ ∧ ∀yψ) ↔ (∀xφ ∧ ∀yψ))
7	3, 6	bitr 151	1 ⊢ (∀x∀y(φ ∧ ψ) ↔ (∀xφ ∧ ∀yψ))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672

This theorem is referenced by:  mo 1020  2eu4 1070

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-7 676  ax-gen 677

This theorem depends on definitions:  df-bi 128  df-an 198

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