Theorem pm4.79 566

Description: Theorem *4.79 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 27-Jun-2013.)

Assertion

Ref	Expression
pm4.79	|- (((ps -> ph) \/ (ch -> ph)) <-> ((ps /\ ch) -> ph))

Proof of Theorem pm4.79

Step	Hyp	Ref	Expression
1		id 19	. . 3 |- ((ps -> ph) -> (ps -> ph))
2		id 19	. . 3 |- ((ch -> ph) -> (ch -> ph))
3	1, 2	jaoa 496	. 2 |- (((ps -> ph) \/ (ch -> ph)) -> ((ps /\ ch) -> ph))
4		simplim 143	. . . 4 |- (-. (ps -> ph) -> ps)
5		pm3.3 431	. . . 4 |- (((ps /\ ch) -> ph) -> (ps -> (ch -> ph)))
6	4, 5	syl5 28	. . 3 |- (((ps /\ ch) -> ph) -> (-. (ps -> ph) -> (ch -> ph)))
7	6	orrd 367	. 2 |- (((ps /\ ch) -> ph) -> ((ps -> ph) \/ (ch -> ph)))
8	3, 7	impbii 180	1 |- (((ps -> ph) \/ (ch -> ph)) <-> ((ps /\ ch) -> ph))

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358

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 177  df-or 359  df-an 360

This theorem is referenced by: (None)