Theorem im3ord 637

Description: Deduction joining 3 implications to form implication of disjunctions.

Hypotheses

Ref	Expression
im3d.1	|- (ph -> (ps -> ch))
im3d.2	|- (ph -> (th -> ta ))
im3d.3	|- (ph -> (et -> ze))

Assertion

Ref	Expression
im3ord	|- (ph -> ((ps \/ th \/ et) -> (ch \/ ta \/ ze)))

Proof of Theorem im3ord

Step	Hyp	Ref	Expression
1		im3d.1	. . . 4 |- (ph -> (ps -> ch))
2		im3d.2	. . . 4 |- (ph -> (th -> ta ))
3	1, 2	orim12d 436	. . 3 |- (ph -> ((ps \/ th) -> (ch \/ ta )))
4		im3d.3	. . 3 |- (ph -> (et -> ze))
5	3, 4	orim12d 436	. 2 |- (ph -> (((ps \/ th) \/ et) -> ((ch \/ ta ) \/ ze)))
6		df-3or 582	. 2 |- ((ps \/ th \/ et) <-> ((ps \/ th) \/ et))
7		df-3or 582	. 2 |- ((ch \/ ta \/ ze) <-> ((ch \/ ta ) \/ ze))
8	5, 6, 7	3imtr4g 426	1 |- (ph -> ((ps \/ th \/ et) -> (ch \/ ta \/ ze)))

Syntax hints:   -> wi 2   \/ wo 195   \/ w3o 580

This theorem is referenced by:  zornlem6 3608

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  df-3or 582

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