Theorem pm4.44 769

Description: Theorem *4.44 of [WhiteheadRussell] p. 119. (Contributed by NM, 3-Jan-2005.)

Assertion

Ref	Expression
pm4.44	⊢ (𝜑 ↔ (𝜑 ∨ (𝜑 ∧ 𝜓)))

Proof of Theorem pm4.44

Step	Hyp	Ref	Expression
1		orc 702	. 2 ⊢ (𝜑 → (𝜑 ∨ (𝜑 ∧ 𝜓)))
2		id 19	. . 3 ⊢ (𝜑 → 𝜑)
3		simpl 108	. . 3 ⊢ ((𝜑 ∧ 𝜓) → 𝜑)
4	2, 3	jaoi 706	. 2 ⊢ ((𝜑 ∨ (𝜑 ∧ 𝜓)) → 𝜑)
5	1, 4	impbii 125	1 ⊢ (𝜑 ↔ (𝜑 ∨ (𝜑 ∧ 𝜓)))

Syntax hints:   ∧ wa 103   ↔ wb 104   ∨ wo 698

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699

This theorem depends on definitions:  df-bi 116

This theorem is referenced by: (None)