Theorem 3jaoi 1303

Description: Disjunction of 3 antecedents (inference). (Contributed by NM, 12-Sep-1995.)

Hypotheses

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

Assertion

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

Proof of Theorem 3jaoi

Step	Hyp	Ref	Expression
1		3jaoi.1	. . 3 |- ( ph -> ps )
2		3jaoi.2	. . 3 |- ( ch -> ps )
3		3jaoi.3	. . 3 |- ( th -> ps )
4	1, 2, 3	3pm3.2i 1175	. 2 |- ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) )
5		3jao 1301	. 2 |- ( ( ( ph -> ps ) /\ ( ch -> ps ) /\ ( th -> ps ) ) -> ( ( ph \/ ch \/ th ) -> ps ) )
6	4, 5	ax-mp 5	1 |- ( ( ph \/ ch \/ th ) -> ps )

Syntax hints:   -> wi 4   \/ w3o 977   /\ w3a 978

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980

This theorem is referenced by:  3jaoian  1305  3ianorr  1309  acexmidlem1  5874  nndceq  6503  nndcel  6504  znegcl  9287  xrltnr  9782  nltpnft  9817  ngtmnft  9820  xrrebnd  9822  xnegcl  9835  xnegneg  9836  xltnegi  9838  xrpnfdc  9845  xrmnfdc  9846  xnegid  9862  xaddid1  9865  xposdif  9885  prm23lt5  12266  zabsle1  14588