Theorem 4exmid 905

Description: The disjunction of the four possible combinations of two wffs and their negations is always true. (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 404	. 2 ⊢ ((φ ↔ ψ) ∨ ¬ (φ ↔ ψ))
2		dfbi3 863	. . 3 ⊢ ((φ ↔ ψ) ↔ ((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)))
3		xor 861	. . 3 ⊢ (¬ (φ ↔ ψ) ↔ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)))
4	2, 3	orbi12i 507	. 2 ⊢ (((φ ↔ ψ) ∨ ¬ (φ ↔ ψ)) ↔ (((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)) ∨ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ))))
5	1, 4	mpbi 199	1 ⊢ (((φ ∧ ψ) ∨ (¬ φ ∧ ¬ ψ)) ∨ ((φ ∧ ¬ ψ) ∨ (ψ ∧ ¬ φ)))

Syntax hints:  ¬ wn 3   ↔ wb 176   ∨ wo 357   ∧ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by: (None)