Theorem 3orim123d 1260

Description: Deduction joining 3 implications to form implication of disjunctions. (Contributed by NM, 4-Apr-1997.)

Hypotheses

Ref	Expression
3anim123d.1	|- (ph -> (ps -> ch))
3anim123d.2	|- (ph -> (th -> ta))
3anim123d.3	|- (ph -> (et -> ze))

Assertion

Ref	Expression
3orim123d	|- (ph -> ((ps \/ th \/ et) -> (ch \/ ta \/ ze)))

Proof of Theorem 3orim123d

Step	Hyp	Ref	Expression
1		3anim123d.1	. . . 4 |- (ph -> (ps -> ch))
2		3anim123d.2	. . . 4 |- (ph -> (th -> ta))
3	1, 2	orim12d 811	. . 3 |- (ph -> ((ps \/ th) -> (ch \/ ta)))
4		3anim123d.3	. . 3 |- (ph -> (et -> ze))
5	3, 4	orim12d 811	. 2 |- (ph -> (((ps \/ th) \/ et) -> ((ch \/ ta) \/ ze)))
6		df-3or 935	. 2 |- ((ps \/ th \/ et) <-> ((ps \/ th) \/ et))
7		df-3or 935	. 2 |- ((ch \/ ta \/ ze) <-> ((ch \/ ta) \/ ze))
8	5, 6, 7	3imtr4g 261	1 |- (ph -> ((ps \/ th \/ et) -> (ch \/ ta \/ ze)))

Syntax hints:   -> wi 4   \/ wo 357   \/ w3o 933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935

This theorem is referenced by:  nncdiv3  6278