Theorem mpjao3dan 1238

Description: Eliminate a 3-way disjunction in a deduction. (Contributed by Thierry Arnoux, 13-Apr-2018.)

Hypotheses

Ref	Expression
mpjao3dan.1	|- ( ( ph /\ ps ) -> ch )
mpjao3dan.2	|- ( ( ph /\ th ) -> ch )
mpjao3dan.3	|- ( ( ph /\ ta ) -> ch )
mpjao3dan.4	|- ( ph -> ( ps \/ th \/ ta ) )

Assertion

Ref	Expression
mpjao3dan	|- ( ph -> ch )

Proof of Theorem mpjao3dan

Step	Hyp	Ref	Expression
1		mpjao3dan.1	. . 3 |- ( ( ph /\ ps ) -> ch )
2		mpjao3dan.2	. . 3 |- ( ( ph /\ th ) -> ch )
3	1, 2	jaodan 743	. 2 |- ( ( ph /\ ( ps \/ th ) ) -> ch )
4		mpjao3dan.3	. 2 |- ( ( ph /\ ta ) -> ch )
5		mpjao3dan.4	. . 3 |- ( ph -> ( ps \/ th \/ ta ) )
6		df-3or 920	. . 3 |- ( ( ps \/ th \/ ta ) <-> ( ( ps \/ th ) \/ ta ) )
7	5, 6	sylib 120	. 2 |- ( ph -> ( ( ps \/ th ) \/ ta ) )
8	3, 4, 7	mpjaodan 744	1 |- ( ph -> ch )

Syntax hints:   -> wi 4   /\ wa 102   \/ wo 661   \/ w3o 918

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662

This theorem depends on definitions:  df-bi 115  df-3or 920

This theorem is referenced by:  wetriext  4319  nntri3  6098  nntri2or2  6099  caucvgprlemnkj  6856  caucvgprlemnbj  6857  caucvgprprlemnkj  6882  caucvgprprlemnbj  6883  caucvgsr  6978  addmodlteq  9400  zdvdsdc  10216  divalglemeunn  10321  divalglemex  10322  divalglemeuneg  10323  divalg  10324  znege1  10556