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Definition df-ifp 1006
Description: Definition of the conditional operator for propositions. The value of if-(𝜑, 𝜓, 𝜒) is 𝜓 if 𝜑 is true and 𝜒 if 𝜑 false. See dfifp2 1007, dfifp3 1008, dfifp4 1009, dfifp5 1010, dfifp6 1011 and dfifp7 1012 for alternate definitions.

This definition (in the form of dfifp2 1007) 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).

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 wff with n+1 variables; single out one variable, say 𝜑; when one sets 𝜑 to True (resp. False), then what remains is a wff of n variables, so by the induction hypothesis it corresponds to a formula using only {if-, ⊤, ⊥}, say 𝜓 (resp. 𝜒); therefore, the formula if-(𝜑, 𝜓, 𝜒) represents the initial wff. Now, since { → , ¬ } and similar systems suffice to express 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 1005 . 2 wff if-(𝜑, 𝜓, 𝜒)
51, 2wa 382 . . 3 wff (𝜑𝜓)
61wn 3 . . . 4 wff ¬ 𝜑
76, 3wa 382 . . 3 wff 𝜑𝜒)
85, 7wo 381 . 2 wff ((𝜑𝜓) ∨ (¬ 𝜑𝜒))
94, 8wb 194 1 wff (if-(𝜑, 𝜓, 𝜒) ↔ ((𝜑𝜓) ∨ (¬ 𝜑𝜒)))
Colors of variables: wff setvar class
This definition is referenced by:  dfifp2  1007  dfifp6  1011  ifpor  1014  casesifp  1019  ifpbi123d  1020  1fpid3  1022  bj-df-ifc  31537  ifpdfan  36628  ifpnot23  36641  1wlk1walk  40841  upgr1wlkwlk  40845
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