Theorem orcana 509

Description: Disjunction in consequent versus conjunction in antecedent. Similar to Theorem *5.6 of [WhiteheadRussell] p. 125.

Assertion

Ref	Expression
orcana	|- ((ph -> (ps \/ ch)) <-> ((ph /\ -. ps) -> ch))

Proof of Theorem orcana

Step	Hyp	Ref	Expression
1		df-or 197	. . 3 |- ((ps \/ ch) <-> (-. ps -> ch))
2	1	imbi2i 160	. 2 |- ((ph -> (ps \/ ch)) <-> (ph -> (-. ps -> ch)))
3		impexp 276	. 2 |- (((ph /\ -. ps) -> ch) <-> (ph -> (-. ps -> ch)))
4	2, 3	bitr4 154	1 |- ((ph -> (ps \/ ch)) <-> ((ph /\ -. ps) -> ch))

Syntax hints:  -. wn 1   -> wi 2   <-> wb 127   \/ wo 195   /\ wa 196

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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