Theorem 3jaodan 1301

Description: Disjunction of 3 antecedents (deduction). (Contributed by NM, 14-Oct-2005.)

Hypotheses

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

Assertion

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

Proof of Theorem 3jaodan

Step	Hyp	Ref	Expression
1		3jaodan.1	. . . 4 |- ( ( ph /\ ps ) -> ch )
2	1	ex 114	. . 3 |- ( ph -> ( ps -> ch ) )
3		3jaodan.2	. . . 4 |- ( ( ph /\ th ) -> ch )
4	3	ex 114	. . 3 |- ( ph -> ( th -> ch ) )
5		3jaodan.3	. . . 4 |- ( ( ph /\ ta ) -> ch )
6	5	ex 114	. . 3 |- ( ph -> ( ta -> ch ) )
7	2, 4, 6	3jaod 1299	. 2 |- ( ph -> ( ( ps \/ th \/ ta ) -> ch ) )
8	7	imp 123	1 |- ( ( ph /\ ( ps \/ th \/ ta ) ) -> ch )

Syntax hints:   -> wi 4   /\ wa 103   \/ 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  df-3an 975

This theorem is referenced by:  zeo  9317  xrltnsym  9750  xrlttr  9752  xrltso  9753  xrlttri3  9754  xltnegi  9792  xaddcom  9818  xnegdi  9825  xsubge0  9838  qbtwnxr  10214  blssioo  13339