Theorem jao 340

Description: Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113.

Assertion

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

Proof of Theorem jao

Step	Hyp	Ref	Expression
1		con3 94	. 2 |- ((ph -> ps) -> (-. ps -> -. ph))
2		pm3.43i 287	. . . . 5 |- ((-. ps -> -. ph) -> ((-. ps -> -. ch) -> (-. ps -> (-. ph /\ -. ch))))
3		con1 92	. . . . 5 |- ((-. ps -> (-. ph /\ -. ch)) -> (-. (-. ph /\ -. ch) -> ps))
4	2, 3	syl6 22	. . . 4 |- ((-. ps -> -. ph) -> ((-. ps -> -. ch) -> (-. (-. ph /\ -. ch) -> ps)))
5		oran 312	. . . 4 |- ((ph \/ ch) <-> -. (-. ph /\ -. ch))
6	4, 5	syl7ib 216	. . 3 |- ((-. ps -> -. ph) -> ((-. ps -> -. ch) -> ((ph \/ ch) -> ps)))
7		con3 94	. . 3 |- ((ch -> ps) -> (-. ps -> -. ch))
8	6, 7	syl5 21	. 2 |- ((-. ps -> -. ph) -> ((ch -> ps) -> ((ph \/ ch) -> ps)))
9	1, 8	syl 10	1 |- ((ph -> ps) -> ((ch -> ps) -> ((ph \/ ch) -> ps)))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223

This theorem is referenced by:  jaoi 341  jaob 422  jaod 424  3jao 886  indpi 5034

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225

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