Theorem looinv 206

Description: The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. Using dfor2 914, we can see that this essentially expresses "disjunction is commutative". Theorem *2.69 of [WhiteheadRussell] p. 108. It is a special instance of the axiom "Roll", see peirceroll 86. (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 84	. 2 ⊢ (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → ((𝜑 → 𝜓) → 𝜑)))
2		peirce 205	. 2 ⊢ (((𝜑 → 𝜓) → 𝜑) → 𝜑)
3	1, 2	syl6 36	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  1765  bj-looinvi  37186