Theorem pm2.5gdc 856

Description: Negating an implication for a decidable antecedent. General instance of Theorem *2.5 of [WhiteheadRussell] p. 107 under a decidability condition. (Contributed by Jim Kingdon, 29-Mar-2018.)

Assertion

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

Proof of Theorem pm2.5gdc

Step	Hyp	Ref	Expression
1		simplimdc 850	. . . 4 |- (DECID ph -> ( -. ( ph -> ps ) -> ph ) )
2	1	imp 123	. . 3 |- ( (DECID ph /\ -. ( ph -> ps ) ) -> ph )
3	2	pm2.24d 612	. 2 |- ( (DECID ph /\ -. ( ph -> ps ) ) -> ( -. ph -> ch ) )
4	3	ex 114	1 |- (DECID ph -> ( -. ( ph -> ps ) -> ( -. ph -> ch ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103  DECID wdc 824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825

This theorem is referenced by:  pm2.5dc  857  pm2.521gdc  858