Theorem looinvdc 821

Description: The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 794, 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 70	. 2 ⊢ (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → ((𝜑 → 𝜓) → 𝜑)))
2		peircedc 820	. 2 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜑) → 𝜑))
3	1, 2	syl9r 67	1 ⊢ (DECID 𝜑 → (((𝜑 → 𝜓) → 𝜓) → ((𝜓 → 𝜑) → 𝜑)))

Syntax hints:   → wi 4  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by: (None)