Theorem 4exmid 977

Description: The disjunction of the four possible combinations of two wffs and their negations is always true. A four-way excluded middle (see exmid 430). (Contributed by David Abernethy, 28-Jan-2014.)

Assertion

Ref	Expression
4exmid	⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))

Proof of Theorem 4exmid

Step	Hyp	Ref	Expression
1		exmid 430	. 2 ⊢ ((𝜑 ↔ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓))
2		dfbi3 933	. . 3 ⊢ ((𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)))
3		xor 931	. . 3 ⊢ (¬ (𝜑 ↔ 𝜓) ↔ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))
4	2, 3	orbi12i 542	. 2 ⊢ (((𝜑 ↔ 𝜓) ∨ ¬ (𝜑 ↔ 𝜓)) ↔ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑))))
5	1, 4	mpbi 219	1 ⊢ (((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ ¬ 𝜓)) ∨ ((𝜑 ∧ ¬ 𝜓) ∨ (𝜓 ∧ ¬ 𝜑)))

Syntax hints:  ¬ wn 3   ↔ wb 195   ∨ wo 382   ∧ wa 383

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 196  df-or 384  df-an 385

This theorem is referenced by:  clsk1indlem3  37361