Theorem prth 429

Description: Theorem *3.47 of [WhiteheadRussell] p. 113. It was proved by Leibniz, and it evidently pleased him enough to call it 'praeclarum theorema.'

Assertion

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

Proof of Theorem prth

Step	Hyp	Ref	Expression
1		pm3.2 232	. . . . 5 |- (ps -> (th -> (ps /\ th)))
2	1	syl3d 26	. . . 4 |- (ps -> ((ch -> th) -> (ch -> (ps /\ th))))
3	2	syl3 18	. . 3 |- ((ph -> ps) -> (ph -> ((ch -> th) -> (ch -> (ps /\ th)))))
4	3	com23 32	. 2 |- ((ph -> ps) -> ((ch -> th) -> (ph -> (ch -> (ps /\ th)))))
5	4	imp4b 283	1 |- (((ph -> ps) /\ (ch -> th)) -> ((ph /\ ch) -> (ps /\ th)))

Syntax hints:   -> wi 2   /\ wa 196

This theorem is referenced by:  anim12d 431  mo 1020  ssxp 2487  tfrlem5 2953  climunii 4883  hlimcaui 5141  hlimunii 5143  spanun 5450

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-an 198

metamath.org