Theorem pm2.5dc 763

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 757	. . . 4 |- (DECID ph -> ( -. ( ph -> ps ) -> ph ) )
2	1	imp 115	. . 3 |- ( (DECID ph /\ -. ( ph -> ps ) ) -> ph )
3	2	pm2.24d 552	. 2 |- ( (DECID ph /\ -. ( ph -> ps ) ) -> ( -. ph -> ps ) )
4	3	ex 108	1 |- (DECID ph -> ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 97  DECID wdc 742

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630

This theorem depends on definitions:  df-bi 110  df-dc 743

This theorem is referenced by:  pm5.11dc  815