Theorem pm2.18 102

Description: Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
pm2.18	|- ((-. ph -> ph) -> ph)

Proof of Theorem pm2.18

Step	Hyp	Ref	Expression
1		pm2.21 100	. . . 4 |- (-. ph -> (ph -> -. (-. ph -> ph)))
2	1	a2i 12	. . 3 |- ((-. ph -> ph) -> (-. ph -> -. (-. ph -> ph)))
3	2	con4d 97	. 2 |- ((-. ph -> ph) -> ((-. ph -> ph) -> ph))
4	3	pm2.43i 43	1 |- ((-. ph -> ph) -> ph)

Syntax hints:  -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem is referenced by:  pm2.18d  103  pm4.81  355  ax10  1944