Theorem pm4.43 856

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		pm3.24 627	. . 3 ⊢ ¬ (𝜓 ∧ ¬ 𝜓)
2	1	biorfi 665	. 2 ⊢ (𝜑 ↔ (𝜑 ∨ (𝜓 ∧ ¬ 𝜓)))
3		ordi 729	. 2 ⊢ ((𝜑 ∨ (𝜓 ∧ ¬ 𝜓)) ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ ¬ 𝜓)))
4	2, 3	bitri 173	1 ⊢ (𝜑 ↔ ((𝜑 ∨ 𝜓) ∧ (𝜑 ∨ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   ∧ wa 97   ↔ wb 98   ∨ wo 629

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110

This theorem is referenced by: (None)