3 \* Intuitionistic Predicate Logic with Equality *\
5 \open elements \* [1] 2.1. 2.2. 3.1 *\
7 \decl "logical false" False: *Prop
9 \decl "logical conjunction" And: { [*Prop] [*Prop] } *Prop
11 \decl "logical disjunction" Or: { [*Prop] [*Prop] } *Prop
13 \* implication and non-dependent abstraction are isomorphic *\
14 \def "logical implication"
15 Imp = { [p:*Prop] [q:*Prop] } [p]^1 q : { [*Prop] [*Prop] } *Prop
17 \* comprehension and dependent abstraction are isomorphic *\
18 \def "unrestricted logical comprehension"
19 All = [q:[*Obj]^1 *Prop] [x:*Obj]^1 q(x) : [[*Obj]^1 *Prop] *Prop
21 \decl "unrestricted logical existence" Ex: [[*Obj]^1 *Prop] *Prop
23 \decl "syntactical identity" Id: { [*Obj] [*Obj] } *Prop
27 \open abbreviations \* [1] 2.3. *\
29 \def "logical negation"
30 Not = [p:*Prop] Imp(p, False) : [*Prop] *Prop
32 \def "logical equivalence"
33 Iff = { [p:*Prop] [q:*Prop] } And(Imp(p, q), Imp(q, p)) : { [*Prop] [*Prop] } *Prop
35 \def "unrestricted strict logical existence"
36 EX = [p:[*Obj]^1 *Prop] Ex([x:*Obj]^2 And(p(x), All([y:*Obj]^2 Imp(p(y), Id(x, y)))))
37 : [[*Obj]^1 *Prop] *Prop
39 \def "negated syntactical identity"
40 NId = { [x:*Obj] [y:*Obj] } Not(Id(x, y)) : { [*Obj] [*Obj] } *Prop