Theorem looinv 174

Description: The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 400, we can see that this essentially expresses "disjunction is commutative". Theorem *2.69 of [WhiteheadRussell] p. 108. (Contributed by NM, 12-Aug-2004.)

Assertion

Ref	Expression
looinv	⊢ (((φ → ψ) → ψ) → ((ψ → φ) → φ))

Proof of Theorem looinv

Step	Hyp	Ref	Expression
1		imim1 70	. 2 ⊢ (((φ → ψ) → ψ) → ((ψ → φ) → ((φ → ψ) → φ)))
2		peirce 172	. 2 ⊢ (((φ → ψ) → φ) → φ)
3	1, 2	syl6 29	1 ⊢ (((φ → ψ) → ψ) → ((ψ → φ) → φ))

This proof depends on syntax axioms:   → wi 4

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is used by:  merco2  1501