Theorem pm2.5dc 796

Description: Negating an implication for a decidable antecedent. Based on theorem *2.5 of [WhiteheadRussell] p. 107. (Contributed by Jim Kingdon, 29-Mar-2018.)

Assertion

Ref	Expression
pm2.5dc	|- (DECID ph -> ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) )

Proof of Theorem pm2.5dc

Step	Hyp	Ref	Expression
1		simplimdc 790	. . . 4 |- (DECID ph -> ( -. ( ph -> ps ) -> ph ) )
2	1	imp 122	. . 3 |- ( (DECID ph /\ -. ( ph -> ps ) ) -> ph )
3	2	pm2.24d 584	. 2 |- ( (DECID ph /\ -. ( ph -> ps ) ) -> ( -. ph -> ps ) )
4	3	ex 113	1 |- (DECID ph -> ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102  DECID wdc 775

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662

This theorem depends on definitions:  df-bi 115  df-dc 776

This theorem is referenced by:  pm5.11dc  848