Theorem annimim 681

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 932. (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 637	. . 3 |- ( ( ( ph -> ps ) -> ps ) -> ( -. ps -> -. ( ph -> ps ) ) )
3	1, 2	syl 14	. 2 |- ( ph -> ( -. ps -> -. ( ph -> ps ) ) )
4	3	imp 123	1 |- ( ( ph /\ -. ps ) -> -. ( ph -> ps ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-in1 609  ax-in2 610

This theorem is referenced by:  pm4.65r  682  imanim  683  pm4.52im  745  dcim  836  exanaliim  1640