Definition df-ifp 984

Description: Definition of the conditional operator for propositions. The expression if- ( ph , ps , ch ) is read "if ph then ps else ch". See dfifp2dc 987, dfifp3dc 988, dfifp4dc 989 and dfifp5dc 990 for alternate definitions.

This definition (in the form of dfifp2dc 987) 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 987 for compatibility with intuitionistic logic development: with this form, it is clear that if- ( ph , ps , ch ) implies decidability of ph (ifpdc 985), 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 983	. 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 713	. 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  985  ifp2  986  dfifp2dc  987  ifpor  993  ifptru  995  ifpfal  996  ifpbi123d  998  1fpid3  1000  wlk1walkdom  16080  upgriswlkdc  16081