Theorem pm4.52 981

Description: Theorem *4.52 of [WhiteheadRussell] p. 120. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 5-Nov-2012.)

Assertion

Ref	Expression
pm4.52	⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∨ 𝜓))

Proof of Theorem pm4.52

Step	Hyp	Ref	Expression
1		annim 403	. 2 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (𝜑 → 𝜓))
2		imor 849	. 2 ⊢ ((𝜑 → 𝜓) ↔ (¬ 𝜑 ∨ 𝜓))
3	1, 2	xchbinx 333	1 ⊢ ((𝜑 ∧ ¬ 𝜓) ↔ ¬ (¬ 𝜑 ∨ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∨ wo 843

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844

This theorem is referenced by:  pm4.53  982  ordtri3  6287  ifpim123g  41005