Definition df-ifp 1058

Description: Definition of the conditional operator for propositions. The expression if-(𝜑, 𝜓, 𝜒) is read "if 𝜑 then 𝜓 else 𝜒". See dfifp2 1059, dfifp3 1060, dfifp4 1061, dfifp5 1062, dfifp6 1063 and dfifp7 1064 for alternate definitions.

This definition (in the form of dfifp2 1059) 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 1059 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 1057	. 2 wff if-(𝜑, 𝜓, 𝜒)
5	1, 2	wa 398	. . 3 wff (𝜑 ∧ 𝜓)
6	1	wn 3	. . . 4 wff ¬ 𝜑
7	6, 3	wa 398	. . 3 wff (¬ 𝜑 ∧ 𝜒)
8	5, 7	wo 843	. 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  1059  dfifp6  1063  ifpor  1066  casesifp  1071  ifpbi123dOLD  1073  1fpid3  1075  wlk1walk  27422  upgriswlk  27424  bj-df-ifc  33915  bj-dfif  33916  bj-ififc  33917  ifpdfan  39838  ifpnot23  39851  upgrwlkupwlk  44022