Definition df-ifp 1069

Description: Definition of the conditional operator for propositions. The expression if-(𝜑, 𝜓, 𝜒) is read "if 𝜑 then 𝜓 else 𝜒". See dfifp2 1070, dfifp3 1071, dfifp4 1072, dfifp5 1073, dfifp6 1074 and dfifp7 1075 for alternate definitions.

This definition (in the form of dfifp2 1070) appears in Section II.24 of [Church] p. 129 (Definition D12 page 132), where it is called "conditioned disjunction". Church's [𝜓, 𝜑, 𝜒] corresponds to our if-(𝜑, 𝜓, 𝜒) (note the permutation of the first two variables).

This form was chosen as the definition rather than dfifp2 1070 for compatibility with intuitionistic logic development: with this form, it is clear that if-(𝜑, 𝜓, 𝜒) implies decidability of 𝜑, which is most often what is wanted.

Church uses the conditional operator as an intermediate step to prove completeness of some systems of connectives. The first result is that the system {if-, ⊤, ⊥} is complete: for the induction step, consider a formula of n+1 variables; single out one variable, say 𝜑; when one sets 𝜑 to True (resp. False), then what remains is a formula of n variables, so by the induction hypothesis it is equivalent to a formula using only the connectives if-, ⊤, ⊥, say 𝜓 (resp. 𝜒); therefore, the formula if-(𝜑, 𝜓, 𝜒) is equivalent to the initial formula of n+1 variables. Now, since { → , ¬ } and similar systems suffice to express the connectives if-, ⊤, ⊥, they are also complete.

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

Assertion

Ref	Expression
df-ifp	⊢ (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))

Detailed syntax breakdown of Definition df-ifp

Step	Hyp	Ref	Expression
1		wph	. . 3 wff 𝜑
2		wps	. . 3 wff 𝜓
3		wch	. . 3 wff 𝜒
4	1, 2, 3	wif 1068	. 2 wff if-(𝜑, 𝜓, 𝜒)
5	1, 2	wa 396	. . 3 wff (𝜑 ∧ 𝜓)
6	1	wn 3	. . . 4 wff ¬ 𝜑
7	6, 3	wa 396	. . 3 wff (¬ 𝜑 ∧ 𝜒)
8	5, 7	wo 853	. 2 wff ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))
9	4, 8	wb 207	1 wff (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))

This definition is referenced by:  dfifp2  1070  dfifp6  1074  ifpor  1078  casesifp  1083  1fpid3  1087  wlk1walk  29732  upgriswlk  29734  bj-df-ifc  36898  bj-dfif  36899  bj-ififc  36900  wl-ifp-ncond1  37833  wl-ifpimpr  37835  ifpdfan  43917  ifpnot23  43929  upgrwlkupwlk  48638