Theorem 3ornot23 1281

Description: If the second and third disjuncts of a true triple disjunction are false, then the first disjunct is true. Automatically derived from 3ornot23VD 16671. (Contributed by Alan Sare, 31-Dec-2011.)

Assertion

Ref	Expression
3ornot23	|- ((-. ph /\ -. ps) -> ((ch \/ ph \/ ps) -> ch))

Proof of Theorem 3ornot23

Step	Hyp	Ref	Expression
1		ioran 331	. . 3 |- (-. (ph \/ ps) <-> (-. ph /\ -. ps))
2	1	biimpri 169	. 2 |- ((-. ph /\ -. ps) -> -. (ph \/ ps))
3		3orass 861	. . . 4 |- ((ch \/ ph \/ ps) <-> (ch \/ (ph \/ ps)))
4	3	biimpi 168	. . 3 |- ((ch \/ ph \/ ps) -> (ch \/ (ph \/ ps)))
5	4	a1i 8	. 2 |- ((-. ph /\ -. ps) -> ((ch \/ ph \/ ps) -> (ch \/ (ph \/ ps))))
6		orel2 272	. 2 |- (-. (ph \/ ps) -> ((ch \/ (ph \/ ps)) -> ch))
7	2, 5, 6	sylsyld 32	1 |- ((-. ph /\ -. ps) -> ((ch \/ ph \/ ps) -> ch))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   \/ w3o 857

This theorem is referenced by:  tratrb 5831  tratrbVD 16685

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859

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