Definition df-ifp 1074

Description: Definition of the conditional operator for propositions. The expression if-(𝜑, 𝜓, 𝜒) is read "if 𝜑 then 𝜓 else 𝜒". See dfifp2 1075, dfifp3 1076, dfifp4 1077, dfifp5 1078, dfifp6 1079 and dfifp7 1080 for alternate definitions.

This definition (in the form of dfifp2 1075) 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 1075 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 1073	. 2 wff if-(𝜑, 𝜓, 𝜒)
5	1, 2	wa 399	. . 3 wff (𝜑 ∧ 𝜓)
6	1	wn 3	. . . 4 wff ¬ 𝜑
7	6, 3	wa 399	. . 3 wff (¬ 𝜑 ∧ 𝜒)
8	5, 7	wo 858	. 2 wff ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒))
9	4, 8	wb 208	1 wff (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑 ∧ 𝜓) ∨ (¬ 𝜑 ∧ 𝜒)))

This definition is referenced by:  dfifp2  1075  dfifp6  1079  ifpor  1083  casesifp  1088  1fpid3  1092  wlk1walk  29785  upgriswlk  29787  bj-df-ifc  36987  bj-dfif  36988  bj-ififc  36989  wl-ifp-ncond1  37922  wl-ifpimpr  37924  ifpdfan  44006  ifpnot23  44018  upgrwlkupwlk  48726