Theorem 3jaod 1392

Description: Disjunction of three antecedents (deduction). (Contributed by NM, 14-Oct-2005.)

Hypotheses

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

Assertion

Ref	Expression
3jaod	|- ( ph -> ( ( ps \/ th \/ ta ) -> ch ) )

Proof of Theorem 3jaod

Step	Hyp	Ref	Expression
1		3jaod.1	. 2 |- ( ph -> ( ps -> ch ) )
2		3jaod.2	. 2 |- ( ph -> ( th -> ch ) )
3		3jaod.3	. 2 |- ( ph -> ( ta -> ch ) )
4		3jao 1389	. 2 |- ( ( ( ps -> ch ) /\ ( th -> ch ) /\ ( ta -> ch ) ) -> ( ( ps \/ th \/ ta ) -> ch ) )
5	1, 2, 3, 4	syl3anc 1326	1 |- ( ph -> ( ( ps \/ th \/ ta ) -> ch ) )

Syntax hints:   -> wi 4   \/ w3o 1036

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 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039

This theorem is referenced by:  3jaodan  1394  3jaao  1396  fntpb  6473  dfwe2  6981  smo11  7461  smoord  7462  omeulem1  7662  omopth2  7664  oaabs2  7725  elfiun  8336  r111  8638  r1pwss  8647  pwcfsdom  9405  winalim2  9518  xmullem  12094  xmulasslem  12115  xlemul1a  12118  xrsupsslem  12137  xrinfmsslem  12138  xrub  12142  ordtbas2  20995  ordtbas  20996  fmfnfmlem4  21761  dyadmbl  23368  scvxcvx  24712  perfectlem2  24955  ostth3  25327  poseq  31750  sltsolem1  31826  lineext  32183  fscgr  32187  colinbtwnle  32225  broutsideof2  32229  lineunray  32254  lineelsb2  32255  elicc3  32311  4atlem11  34895  dalawlem10  35166  frege129d  38055  goldbachth  41459  perfectALTVlem2  41631