Theorem 3orim123d 1315

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 781	. . 3 |- ( ph -> ( ( ps \/ th ) -> ( ch \/ ta ) ) )
4		3anim123d.3	. . 3 |- ( ph -> ( et -> ze ) )
5	3, 4	orim12d 781	. 2 |- ( ph -> ( ( ( ps \/ th ) \/ et ) -> ( ( ch \/ ta ) \/ ze ) ) )
6		df-3or 974	. 2 |- ( ( ps \/ th \/ et ) <-> ( ( ps \/ th ) \/ et ) )
7		df-3or 974	. 2 |- ( ( ch \/ ta \/ ze ) <-> ( ( ch \/ ta ) \/ ze ) )
8	5, 6, 7	3imtr4g 204	1 |- ( ph -> ( ( ps \/ th \/ et ) -> ( ch \/ ta \/ ze ) ) )

Syntax hints:   -> wi 4   \/ wo 703   \/ w3o 972

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-io 704

This theorem depends on definitions:  df-bi 116  df-3or 974

This theorem is referenced by:  ztri3or0  9254