Theorem looinv 193

Description: The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 426, we can see that this essentially expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108. It is a special instance of the axiom "Roll", see peirceroll 83. (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 81	. 2 ⊢ (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → ((𝜑 → 𝜓) → 𝜑)))
2		peirce 192	. 2 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
3	1, 2	syl6 34	1 ⊢ (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → 𝜑))

Syntax hints:   → wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  merco2  1652  bj-looinvi  31724