Theorem looinvdc 910

Description: The Inversion Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, but where one of the propositions is decidable. Using dfor2dc 890, 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 76	. 2 |- ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ( ( ph -> ps ) -> ph ) ) )
2		peircedc 909	. 2 |- (DECID ph -> ( ( ( ph -> ps ) -> ph ) -> ph ) )
3	1, 2	syl9r 73	1 |- (DECID ph -> ( ( ( ph -> ps ) -> ps ) -> ( ( ps -> ph ) -> ph ) ) )

Syntax hints:   -> wi 4  DECID wdc 829

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 610  ax-io 704

This theorem depends on definitions:  df-bi 116  df-dc 830

This theorem is referenced by: (None)