Theorem pm5.11dc 815

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 784	. 2 |- (DECID ph -> (DECID ps -> DECID ( ph -> ps ) ) )
2		pm2.5dc 763	. . 3 |- (DECID ph -> ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) )
3		pm2.54dc 790	. . 3 |- (DECID ( ph -> ps ) -> ( ( -. ( ph -> ps ) -> ( -. ph -> ps ) ) -> ( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) )
4	2, 3	syl5com 26	. 2 |- (DECID ph -> (DECID ( ph -> ps ) -> ( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) )
5	1, 4	syld 40	1 |- (DECID ph -> (DECID ps -> ( ( ph -> ps ) \/ ( -. ph -> ps ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 629  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: (None)