Theorem pm4.42 1059

Description: Theorem *4.42 of [WhiteheadRussell] p. 119. See also ifpid 1082. (Contributed by Roy F. Longton, 21-Jun-2005.)

Assertion

Ref	Expression
pm4.42	⊢ (𝜑 ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))

Proof of Theorem pm4.42

Step	Hyp	Ref	Expression
1		dedlema 1051	. 2 ⊢ (𝜓 → (𝜑 ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓))))
2		dedlemb 1052	. 2 ⊢ (¬ 𝜓 → (𝜑 ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓))))
3	1, 2	pm2.61i 183	1 ⊢ (𝜑 ↔ ((𝜑 ∧ 𝜓) ∨ (𝜑 ∧ ¬ 𝜓)))

Syntax hints:  ¬ wn 3   ↔ wb 207   ∧ wa 396   ∨ wo 853

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 208  df-an 397  df-or 854

This theorem is referenced by:  inundif  4414  elim2ifim  32640  smatrcl  33987  expdioph  43475