Theorem dfifp2dc 987

Description: Alternate definition of the conditional operator for decidable propositions. The value of if- ( ph , ps , ch ) is "if ph then ps, and if not ph then ch". This is the definition used in Section II.24 of [Church] p. 129 (Definition D12 page 132) (see comment of df-ifp 984). (Contributed by BJ, 22-Jun-2019.)

Assertion

Ref	Expression
dfifp2dc	|- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) )

Proof of Theorem dfifp2dc

Step	Hyp	Ref	Expression
1		ifp2 986	. 2 |- (if- ( ph , ps , ch ) -> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) )
2		exmiddc 841	. . . . 5 |- (DECID ph -> ( ph \/ -. ph ) )
3		simpl 109	. . . . . . . 8 |- ( ( ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ph )
4		simprl 529	. . . . . . . 8 |- ( ( ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( ph -> ps ) )
5	3, 4	jcai 311	. . . . . . 7 |- ( ( ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( ph /\ ps ) )
6	5	orcd 738	. . . . . 6 |- ( ( ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
7		simpl 109	. . . . . . . 8 |- ( ( -. ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> -. ph )
8		simprr 531	. . . . . . . 8 |- ( ( -. ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( -. ph -> ch ) )
9	7, 8	jcai 311	. . . . . . 7 |- ( ( -. ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( -. ph /\ ch ) )
10	9	olcd 739	. . . . . 6 |- ( ( -. ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
11	6, 10	jaoian 800	. . . . 5 |- ( ( ( ph \/ -. ph ) /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
12	2, 11	sylan 283	. . . 4 |- ( (DECID ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
13		df-ifp 984	. . . 4 |- (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )
14	12, 13	sylibr 134	. . 3 |- ( (DECID ph /\ ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) -> if- ( ph , ps , ch ) )
15	14	ex 115	. 2 |- (DECID ph -> ( ( ( ph -> ps ) /\ ( -. ph -> ch ) ) -> if- ( ph , ps , ch ) ) )
16	1, 15	impbid2 143	1 |- (DECID ph -> (if- ( ph , ps , ch ) <-> ( ( ph -> ps ) /\ ( -. ph -> ch ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 713  DECID wdc 839  if-wif 983

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in2 618  ax-io 714

This theorem depends on definitions:  df-bi 117  df-dc 840  df-ifp 984

This theorem is referenced by:  dfifp3dc  988  dfifp5dc  990  ifpdfbidc  991