Theorem pm4.43 1019

Description: Theorem *4.43 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 26-Nov-2012.)

Assertion

Ref	Expression
pm4.43	⊢ (𝜑 ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ ¬ 𝜓)))

Proof of Theorem pm4.43

Step	Hyp	Ref	Expression
1		pm3.24 402	. . 3 ⊢ ¬ (𝜓 ∧ ¬ 𝜓)
2	1	biorfi 935	. 2 ⊢ (𝜑 ↔ (𝜑 ∨ (𝜓 ∧ ¬ 𝜓)))
3		ordi 1002	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ ¬ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ ¬ 𝜓)))
4	2, 3	bitri 274	1 ⊢ (𝜑 ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ 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:  stoweidlem26  43457