Theorem anor 252

Description: Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120.

Assertion

Ref	Expression
anor	⊢ ((φ ∧ ψ) ↔ ¬ (¬ φ ∨ ¬ ψ))

Proof of Theorem anor

Step	Hyp	Ref	Expression
1		df-an 198	. 2 ⊢ ((φ ∧ ψ) ↔ ¬ (φ → ¬ ψ))
2		imor 204	. . 3 ⊢ ((φ → ¬ ψ) ↔ (¬ φ ∨ ¬ ψ))
3	2	negbii 162	. 2 ⊢ (¬ (φ → ¬ ψ) ↔ ¬ (¬ φ ∨ ¬ ψ))
4	1, 3	bitr 151	1 ⊢ ((φ ∧ ψ) ↔ ¬ (¬ φ ∨ ¬ ψ))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196

This theorem is referenced by:  ianor 253  ioran 254  andi 456

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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