Definition df-ifp 987

Description: Definition of the conditional operator for propositions. The expression if- ( ph , ps , ch ) is read "if ph then ps else ch". See dfifp2dc 990, dfifp3dc 991, dfifp4dc 992 and dfifp5dc 993 for alternate definitions.

This definition (in the form of dfifp2dc 990) appears in Section II.24 of [Church] p. 129 (Definition D12 page 132), where it is called "conditioned disjunction". Church's [ ps , ph , ch ] corresponds to our if- ( ph , ps , ch ) (note the permutation of the first two variables).

This form was chosen as the definition rather than dfifp2dc 990 for compatibility with intuitionistic logic development: with this form, it is clear that if- ( ph , ps , ch ) implies decidability of ph (ifpdc 988), which is most often what is wanted.

(Contributed by BJ, 22-Jun-2019.)

Assertion

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

Detailed syntax breakdown of Definition df-ifp

Step	Hyp	Ref	Expression
1		wph	. . 3 wff ph
2		wps	. . 3 wff ps
3		wch	. . 3 wff ch
4	1, 2, 3	wif 986	. 2 wff if- ( ph , ps , ch )
5	1, 2	wa 104	. . 3 wff ( ph /\ ps )
6	1	wn 3	. . . 4 wff -. ph
7	6, 3	wa 104	. . 3 wff ( -. ph /\ ch )
8	5, 7	wo 716	. 2 wff ( ( ph /\ ps ) \/ ( -. ph /\ ch ) )
9	4, 8	wb 105	1 wff (if- ( ph , ps , ch ) <-> ( ( ph /\ ps ) \/ ( -. ph /\ ch ) ) )

This definition is referenced by:  ifpdc  988  ifp2  989  dfifp2dc  990  ifpor  996  ifptru  998  ifpfal  999  ifpbi123d  1001  1fpid3  1003  wlk1walkdom  16280  upgriswlkdc  16281