Theorem looinvdc 905

Description: The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 885, we can see that this expresses "disjunction commutes." Theorem *2.69 of [WhiteheadRussell] p. 108 (plus the decidability condition). (Contributed by NM, 12-Aug-2004.)

Assertion

Ref	Expression
looinvdc	⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → 𝜑)))

Proof of Theorem looinvdc

Step	Hyp	Ref	Expression
1		imim1 76	. 2 ⊢ (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → ((𝜑 → 𝜓) → 𝜑)))
2		peircedc 904	. 2 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
3	1, 2	syl9r 73	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → 𝜑)))

Syntax hints:   → wi 4  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-dc 825

This theorem is referenced by: (None)