Theorem pm5.11dc 899

Description: A decidable proposition or its negation implies a second proposition. Based on theorem *5.11 of [WhiteheadRussell] p. 123. (Contributed by Jim Kingdon, 29-Mar-2018.)

Assertion

Ref	Expression
pm5.11dc	|- (DECID ph -> (DECID ps -> ( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) )

Proof of Theorem pm5.11dc

Step	Hyp	Ref	Expression
1		dcim 831	. 2 |- (DECID ph -> (DECID ps -> DECID ( ph -> ps ) ) )
2		pm2.5dc 857	. . 3 |- (DECID ph -> ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) )
3		pm2.54dc 881	. . 3 |- (DECID ( ph -> ps ) -> ( ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) -> ( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) )
4	2, 3	syl5com 29	. 2 |- (DECID ph -> (DECID ( ph -> ps ) -> ( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) )
5	1, 4	syld 45	1 |- (DECID ph -> (DECID ps -> ( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 698  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: (None)