Theorem pm2.65 649

Description: Theorem *2.65 of [WhiteheadRussell] p. 107. Proof by contradiction. Proofs, such as this one, which assume a proposition, here ph, derive a contradiction, and therefore conclude -. ph, are valid intuitionistically (and can be called "proof of negation", for example by Section 1.2 of [Bauer] p. 482). By contrast, proofs which assume -. ph, derive a contradiction, and conclude ph, such as condandc 871, are only valid for decidable propositions. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 8-Mar-2013.)

Assertion

Ref	Expression
pm2.65	|- ( ( ph -> ps ) -> ( ( ph -> -. ps ) -> -. ph ) )

Proof of Theorem pm2.65

Step	Hyp	Ref	Expression
1		pm2.27 40	. . . 4 |- ( ph -> ( ( ph -> -. ps ) -> -. ps ) )
2	1	con2d 614	. . 3 |- ( ph -> ( ps -> -. ( ph -> -. ps ) ) )
3	2	a2i 11	. 2 |- ( ( ph -> ps ) -> ( ph -> -. ( ph -> -. ps ) ) )
4	3	con2d 614	1 |- ( ( ph -> ps ) -> ( ( ph -> -. ps ) -> -. ph ) )

Syntax hints:   -. wn 3   -> wi 4

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-in1 604  ax-in2 605

This theorem is referenced by:  pm4.82  940