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 ph -> ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ph ) ) )

Proof of Theorem looinvdc

Step	Hyp	Ref	Expression
1		imim1 70	. 2 |- ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ( ( ph -> ps ) -> ph ) ) )
2		peircedc 820	. 2 |- (DECID ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )
3	1, 2	syl9r 67	1 |- (DECID ph -> ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ph ) ) )

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)