Theorem jaob 786

Description: Disjunction of antecedents. Compare Theorem *4.77 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-May-1994.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)

Assertion

Ref	Expression
jaob	|- ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) )

Proof of Theorem jaob

Step	Hyp	Ref	Expression
1		pm2.67-2 402	. . 3 |- ( ( ( ph \/ ch ) -> ps ) -> ( ph -> ps ) )
2		olc 384	. . . 4 |- ( ch -> ( ph \/ ch ) )
3	2	imim1i 59	. . 3 |- ( ( ( ph \/ ch ) -> ps ) -> ( ch -> ps ) )
4	1, 3	jca 532	. 2 |- ( ( ( ph \/ ch ) -> ps ) -> ( ( ph -> ps ) /\ ( ch -> ps ) ) )
5		pm3.44 511	. 2 |- ( ( ( ph -> ps ) /\ ( ch -> ps ) ) -> ( ( ph \/ ch ) -> ps ) )
6	4, 5	impbii 189	1 |- ( ( ( ph \/ ch ) -> ps ) <-> ( ( ph -> ps ) /\ ( ch -> ps ) ) )

Syntax hints:   -> wi 4   <-> wb 186   \/ wo 368   /\ wa 369

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 187  df-or 370  df-an 371

This theorem is referenced by:  pm4.77  790  pm5.53  799  pm4.83  932  axio  2372  unss  3619  ralunb  3626  intun  4262  intpr  4263  relop  4976  sqrt2irr  14193  algcvgblem  14417  efgred  17092  caucfil  22016  plydivex  22987  2sqlem6  24027  arg-ax  30661  tendoeq2  33806  ifpidg  35595