Theorem jao 274

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 86	. 2 |- ((ph -> ps) -> (-. ps -> -. ph))
2		pm3.43i 235	. . . . 5 |- ((-. ps -> -. ph) -> ((-. ps -> -. ch) -> (-. ps -> (-. ph /\ -. ch))))
3		con1 84	. . . . 5 |- ((-. ps -> (-. ph /\ -. ch)) -> (-. (-. ph /\ -. ch) -> ps))
4	2, 3	syl6 23	. . . 4 |- ((-. ps -> -. ph) -> ((-. ps -> -. ch) -> (-. (-. ph /\ -. ch) -> ps)))
5		oran 255	. . . 4 |- ((ph \/ ch) <-> -. (-. ph /\ -. ch))
6	4, 5	bisyl7 189	. . 3 |- ((-. ps -> -. ph) -> ((-. ps -> -. ch) -> ((ph \/ ch) -> ps)))
7		con3 86	. . 3 |- ((ch -> ps) -> (-. ps -> -. ch))
8	6, 7	syl5 22	. 2 |- ((-. ps -> -. ph) -> ((ch -> ps) -> ((ph \/ ch) -> ps)))
9	1, 8	syl 12	1 |- ((ph -> ps) -> ((ch -> ps) -> ((ph \/ ch) -> ps)))

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196

This theorem is referenced by:  jaoi 275  jaob 328  jaod 329  3jao 632  indpi 3828

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

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198

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