Theorem annimim 687

Description: Express conjunction in terms of implication. One direction of Theorem *4.61 of [WhiteheadRussell] p. 120. The converse holds for decidable propositions, as can be seen at annimdc 939. (Contributed by Jim Kingdon, 24-Dec-2017.)

Assertion

Ref	Expression
annimim	|- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )

Proof of Theorem annimim

Step	Hyp	Ref	Expression
1		pm2.27 40	. . 3 |- ( ph -> ( ( ph -> ps ) -> ps ) )
2		con3 643	. . 3 |- ( ( ( ph -> ps ) -> ps ) -> ( -. ps -> -. ( ph -> ps ) ) )
3	1, 2	syl 14	. 2 |- ( ph -> ( -. ps -> -. ( ph -> ps ) ) )
4	3	imp 124	1 |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-in1 615  ax-in2 616

This theorem is referenced by:  pm4.65r  688  imanim  689  pm4.52im  751  dcim  842  exanaliim  1658