Theorem pm3.48 430

Description: Theorem *3.48 of [WhiteheadRussell] p. 114.

Assertion

Ref	Expression
pm3.48	|- (((ph -> ps) /\ (ch -> th)) -> ((ph \/ ch) -> (ps \/ th)))

Proof of Theorem pm3.48

Step	Hyp	Ref	Expression
1		pm3.26 256	. . . 4 |- (((ph -> ps) /\ (ch -> th)) -> (ph -> ps))
2	1	con3d 87	. . 3 |- (((ph -> ps) /\ (ch -> th)) -> (-. ps -> -. ph))
3		pm3.27 260	. . 3 |- (((ph -> ps) /\ (ch -> th)) -> (ch -> th))
4	2, 3	syl34d 29	. 2 |- (((ph -> ps) /\ (ch -> th)) -> ((-. ph -> ch) -> (-. ps -> th)))
5		df-or 197	. 2 |- ((ph \/ ch) <-> (-. ph -> ch))
6		df-or 197	. 2 |- ((ps \/ th) <-> (-. ps -> th))
7	4, 5, 6	3imtr4g 426	1 |- (((ph -> ps) /\ (ch -> th)) -> ((ph \/ ch) -> (ps \/ th)))

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

This theorem is referenced by:  orim12d 436  tz7.48lem 2993

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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