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Definition df-ifp 1055
Description: Definition of the conditional operator for propositions. The expression if-(𝜑, 𝜓, 𝜒) is read "if 𝜑 then 𝜓 else 𝜒". See dfifp2 1056, dfifp3 1057, dfifp4 1058, dfifp5 1059, dfifp6 1060 and dfifp7 1061 for alternate definitions.

This definition (in the form of dfifp2 1056) 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 1056 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) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))

Detailed syntax breakdown of Definition df-ifp
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 wps . . 3 wff 𝜓
3 wch . . 3 wff 𝜒
41, 2, 3wif 1054 . 2 wff if-(𝜑, 𝜓, 𝜒)
51, 2wa 396 . . 3 wff (𝜑𝜓)
61wn 3 . . . 4 wff ¬ 𝜑
76, 3wa 396 . . 3 wff 𝜑𝜒)
85, 7wo 841 . 2 wff ((𝜑𝜓) ∨ (¬ 𝜑𝜒))
94, 8wb 207 1 wff (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
Colors of variables: wff setvar class
This definition is referenced by:  dfifp2  1056  dfifp6  1060  ifpor  1063  casesifp  1068  ifpbi123d  1069  1fpid3  1071  wlk1walk  27347  upgriswlk  27349  bj-df-ifc  33810  bj-dfif  33811  bj-ififc  33812  ifpdfan  39709  ifpnot23  39722  upgrwlkupwlk  43892
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