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Theorem List for Intuitionistic Logic Explorer - 3301-3400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrabeq0 3301 Condition for a restricted class abstraction to be empty. (Contributed by Jeff Madsen, 7-Jun-2010.)
({𝑥𝐴𝜑} = ∅ ↔ ∀𝑥𝐴 ¬ 𝜑)
 
Theoremabeq0 3302 Condition for a class abstraction to be empty. (Contributed by Jim Kingdon, 12-Aug-2018.)
({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
 
Theoremrabxmdc 3303* Law of excluded middle given decidability, in terms of restricted class abstractions. (Contributed by Jim Kingdon, 2-Aug-2018.)
(∀𝑥DECID 𝜑𝐴 = ({𝑥𝐴𝜑} ∪ {𝑥𝐴 ∣ ¬ 𝜑}))
 
Theoremrabnc 3304* Law of noncontradiction, in terms of restricted class abstractions. (Contributed by Jeff Madsen, 20-Jun-2011.)
({𝑥𝐴𝜑} ∩ {𝑥𝐴 ∣ ¬ 𝜑}) = ∅
 
Theoremun0 3305 The union of a class with the empty set is itself. Theorem 24 of [Suppes] p. 27. (Contributed by NM, 5-Aug-1993.)
(𝐴 ∪ ∅) = 𝐴
 
Theoremin0 3306 The intersection of a class with the empty set is the empty set. Theorem 16 of [Suppes] p. 26. (Contributed by NM, 5-Aug-1993.)
(𝐴 ∩ ∅) = ∅
 
Theoreminv1 3307 The intersection of a class with the universal class is itself. Exercise 4.10(k) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(𝐴 ∩ V) = 𝐴
 
Theoremunv 3308 The union of a class with the universal class is the universal class. Exercise 4.10(l) of [Mendelson] p. 231. (Contributed by NM, 17-May-1998.)
(𝐴 ∪ V) = V
 
Theorem0ss 3309 The null set is a subset of any class. Part of Exercise 1 of [TakeutiZaring] p. 22. (Contributed by NM, 5-Aug-1993.)
∅ ⊆ 𝐴
 
Theoremss0b 3310 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23 and its converse. (Contributed by NM, 17-Sep-2003.)
(𝐴 ⊆ ∅ ↔ 𝐴 = ∅)
 
Theoremss0 3311 Any subset of the empty set is empty. Theorem 5 of [Suppes] p. 23. (Contributed by NM, 13-Aug-1994.)
(𝐴 ⊆ ∅ → 𝐴 = ∅)
 
Theoremsseq0 3312 A subclass of an empty class is empty. (Contributed by NM, 7-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵𝐵 = ∅) → 𝐴 = ∅)
 
Theoremssn0 3313 A class with a nonempty subclass is nonempty. (Contributed by NM, 17-Feb-2007.)
((𝐴𝐵𝐴 ≠ ∅) → 𝐵 ≠ ∅)
 
Theoremabf 3314 A class builder with a false argument is empty. (Contributed by NM, 20-Jan-2012.)
¬ 𝜑       {𝑥𝜑} = ∅
 
Theoremeq0rdv 3315* Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
(𝜑 → ¬ 𝑥𝐴)       (𝜑𝐴 = ∅)
 
Theoremcsbprc 3316 The proper substitution of a proper class for a set into a class results in the empty set. (Contributed by NM, 17-Aug-2018.)
𝐴 ∈ V → 𝐴 / 𝑥𝐵 = ∅)
 
Theoremun00 3317 Two classes are empty iff their union is empty. (Contributed by NM, 11-Aug-2004.)
((𝐴 = ∅ ∧ 𝐵 = ∅) ↔ (𝐴𝐵) = ∅)
 
Theoremvss 3318 Only the universal class has the universal class as a subclass. (Contributed by NM, 17-Sep-2003.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(V ⊆ 𝐴𝐴 = V)
 
Theoremdisj 3319* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥𝐵)
 
Theoremdisjr 3320* Two ways of saying that two classes are disjoint. (Contributed by Jeff Madsen, 19-Jun-2011.)
((𝐴𝐵) = ∅ ↔ ∀𝑥𝐵 ¬ 𝑥𝐴)
 
Theoremdisj1 3321* Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 19-Aug-1993.)
((𝐴𝐵) = ∅ ↔ ∀𝑥(𝑥𝐴 → ¬ 𝑥𝐵))
 
Theoremreldisj 3322 Two ways of saying that two classes are disjoint, using the complement of 𝐵 relative to a universe 𝐶. (Contributed by NM, 15-Feb-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴𝐶 → ((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (𝐶𝐵)))
 
Theoremdisj3 3323 Two ways of saying that two classes are disjoint. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 = (𝐴𝐵))
 
Theoremdisjne 3324 Members of disjoint sets are not equal. (Contributed by NM, 28-Mar-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴𝐷𝐵) → 𝐶𝐷)
 
Theoremdisjel 3325 A set can't belong to both members of disjoint classes. (Contributed by NM, 28-Feb-2015.)
(((𝐴𝐵) = ∅ ∧ 𝐶𝐴) → ¬ 𝐶𝐵)
 
Theoremdisj2 3326 Two ways of saying that two classes are disjoint. (Contributed by NM, 17-May-1998.)
((𝐴𝐵) = ∅ ↔ 𝐴 ⊆ (V ∖ 𝐵))
 
Theoremssdisj 3327 Intersection with a subclass of a disjoint class. (Contributed by FL, 24-Jan-2007.)
((𝐴𝐵 ∧ (𝐵𝐶) = ∅) → (𝐴𝐶) = ∅)
 
Theoremundisj1 3328 The union of disjoint classes is disjoint. (Contributed by NM, 26-Sep-2004.)
(((𝐴𝐶) = ∅ ∧ (𝐵𝐶) = ∅) ↔ ((𝐴𝐵) ∩ 𝐶) = ∅)
 
Theoremundisj2 3329 The union of disjoint classes is disjoint. (Contributed by NM, 13-Sep-2004.)
(((𝐴𝐵) = ∅ ∧ (𝐴𝐶) = ∅) ↔ (𝐴 ∩ (𝐵𝐶)) = ∅)
 
Theoremssindif0im 3330 Subclass implies empty intersection with difference from the universal class. (Contributed by NM, 17-Sep-2003.)
(𝐴𝐵 → (𝐴 ∩ (V ∖ 𝐵)) = ∅)
 
Theoreminelcm 3331 The intersection of classes with a common member is nonempty. (Contributed by NM, 7-Apr-1994.)
((𝐴𝐵𝐴𝐶) → (𝐵𝐶) ≠ ∅)
 
Theoremminel 3332 A minimum element of a class has no elements in common with the class. (Contributed by NM, 22-Jun-1994.)
((𝐴𝐵 ∧ (𝐶𝐵) = ∅) → ¬ 𝐴𝐶)
 
Theoremundif4 3333 Distribute union over difference. (Contributed by NM, 17-May-1998.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐶) = ∅ → (𝐴 ∪ (𝐵𝐶)) = ((𝐴𝐵) ∖ 𝐶))
 
Theoremdisjssun 3334 Subset relation for disjoint classes. (Contributed by NM, 25-Oct-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) = ∅ → (𝐴 ⊆ (𝐵𝐶) ↔ 𝐴𝐶))
 
Theoremssdif0im 3335 Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)
(𝐴𝐵 → (𝐴𝐵) = ∅)
 
Theoremvdif0im 3336 Universal class equality in terms of empty difference. (Contributed by Jim Kingdon, 3-Aug-2018.)
(𝐴 = V → (V ∖ 𝐴) = ∅)
 
Theoremdifrab0eqim 3337* If the difference between the restricting class of a restricted class abstraction and the restricted class abstraction is empty, the restricting class is equal to this restricted class abstraction. (Contributed by Jim Kingdon, 3-Aug-2018.)
(𝑉 = {𝑥𝑉𝜑} → (𝑉 ∖ {𝑥𝑉𝜑}) = ∅)
 
Theoreminssdif0im 3338 Intersection, subclass, and difference relationship. In classical logic the converse would also hold. (Contributed by Jim Kingdon, 3-Aug-2018.)
((𝐴𝐵) ⊆ 𝐶 → (𝐴 ∩ (𝐵𝐶)) = ∅)
 
Theoremdifid 3339 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. (Contributed by NM, 22-Apr-2004.)
(𝐴𝐴) = ∅
 
TheoremdifidALT 3340 The difference between a class and itself is the empty set. Proposition 5.15 of [TakeutiZaring] p. 20. Also Theorem 32 of [Suppes] p. 28. Alternate proof of difid 3339. (Contributed by David Abernethy, 17-Jun-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝐴𝐴) = ∅
 
Theoremdif0 3341 The difference between a class and the empty set. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
(𝐴 ∖ ∅) = 𝐴
 
Theorem0dif 3342 The difference between the empty set and a class. Part of Exercise 4.4 of [Stoll] p. 16. (Contributed by NM, 17-Aug-2004.)
(∅ ∖ 𝐴) = ∅
 
Theoremdisjdif 3343 A class and its relative complement are disjoint. Theorem 38 of [Suppes] p. 29. (Contributed by NM, 24-Mar-1998.)
(𝐴 ∩ (𝐵𝐴)) = ∅
 
Theoremdifin0 3344 The difference of a class from its intersection is empty. Theorem 37 of [Suppes] p. 29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∖ 𝐵) = ∅
 
Theoremundif1ss 3345 Absorption of difference by union. In classical logic, as Theorem 35 of [Suppes] p. 29, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
((𝐴𝐵) ∪ 𝐵) ⊆ (𝐴𝐵)
 
Theoremundif2ss 3346 Absorption of difference by union. In classical logic, as in Part of proof of Corollary 6K of [Enderton] p. 144, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
(𝐴 ∪ (𝐵𝐴)) ⊆ (𝐴𝐵)
 
Theoremundifabs 3347 Absorption of difference by union. (Contributed by NM, 18-Aug-2013.)
(𝐴 ∪ (𝐴𝐵)) = 𝐴
 
Theoreminundifss 3348 The intersection and class difference of a class with another class are contained in the original class. In classical logic we'd be able to make a stronger statement: that everything in the original class is in the intersection or the difference (that is, this theorem would be equality rather than subset). (Contributed by Jim Kingdon, 4-Aug-2018.)
((𝐴𝐵) ∪ (𝐴𝐵)) ⊆ 𝐴
 
Theoremdifun2 3349 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)
 
Theoremundifss 3350 Union of complementary parts into whole. (Contributed by Jim Kingdon, 4-Aug-2018.)
(𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) ⊆ 𝐵)
 
Theoremssdifin0 3351 A subset of a difference does not intersect the subtrahend. (Contributed by Jeff Hankins, 1-Sep-2013.) (Proof shortened by Mario Carneiro, 24-Aug-2015.)
(𝐴 ⊆ (𝐵𝐶) → (𝐴𝐶) = ∅)
 
Theoremssdifeq0 3352 A class is a subclass of itself subtracted from another iff it is the empty set. (Contributed by Steve Rodriguez, 20-Nov-2015.)
(𝐴 ⊆ (𝐵𝐴) ↔ 𝐴 = ∅)
 
Theoremssundifim 3353 A consequence of inclusion in the union of two classes. In classical logic this would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
(𝐴 ⊆ (𝐵𝐶) → (𝐴𝐵) ⊆ 𝐶)
 
Theoremdifdifdirss 3354 Distributive law for class difference. In classical logic, as in Exercise 4.8 of [Stoll] p. 16, this would be equality rather than subset. (Contributed by Jim Kingdon, 4-Aug-2018.)
((𝐴𝐵) ∖ 𝐶) ⊆ ((𝐴𝐶) ∖ (𝐵𝐶))
 
Theoremuneqdifeqim 3355 Two ways that 𝐴 and 𝐵 can "partition" 𝐶 (when 𝐴 and 𝐵 don't overlap and 𝐴 is a part of 𝐶). In classical logic, the second implication would be a biconditional. (Contributed by Jim Kingdon, 4-Aug-2018.)
((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 → (𝐶𝐴) = 𝐵))
 
Theoremr19.2m 3356* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1572). The restricted version is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)
 
Theoremr19.3rm 3357* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 19-Dec-2018.)
𝑥𝜑       (∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 
Theoremr19.28m 3358* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
𝑥𝜑       (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
 
Theoremr19.3rmv 3359* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 6-Aug-2018.)
(∃𝑦 𝑦𝐴 → (𝜑 ↔ ∀𝑥𝐴 𝜑))
 
Theoremr19.9rmv 3360* Restricted quantification of wff not containing quantified variable. (Contributed by Jim Kingdon, 5-Aug-2018.)
(∃𝑦 𝑦𝐴 → (𝜑 ↔ ∃𝑥𝐴 𝜑))
 
Theoremr19.28mv 3361* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
(∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))
 
Theoremr19.45mv 3362* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(∃𝑥 𝑥𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝐴 𝜓)))
 
Theoremr19.44mv 3363* Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(∃𝑦 𝑦𝐴 → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))
 
Theoremr19.27m 3364* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
𝑥𝜓       (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
 
Theoremr19.27mv 3365* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is inhabited. (Contributed by Jim Kingdon, 5-Aug-2018.)
(∃𝑥 𝑥𝐴 → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))
 
Theoremrzal 3366* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = ∅ → ∀𝑥𝐴 𝜑)
 
Theoremrexn0 3367* Restricted existential quantification implies its restriction is nonempty (it is also inhabited as shown in rexm 3368). (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
(∃𝑥𝐴 𝜑𝐴 ≠ ∅)
 
Theoremrexm 3368* Restricted existential quantification implies its restriction is inhabited. (Contributed by Jim Kingdon, 16-Oct-2018.)
(∃𝑥𝐴 𝜑 → ∃𝑥 𝑥𝐴)
 
Theoremralidm 3369* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
(∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
 
Theoremral0 3370 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝜑
 
Theoremrgenm 3371* Generalization rule that eliminates an inhabited class requirement. (Contributed by Jim Kingdon, 5-Aug-2018.)
((∃𝑥 𝑥𝐴𝑥𝐴) → 𝜑)       𝑥𝐴 𝜑
 
Theoremralf0 3372* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
¬ 𝜑       (∀𝑥𝐴 𝜑𝐴 = ∅)
 
Theoremralm 3373 Inhabited classes and restricted quantification. (Contributed by Jim Kingdon, 6-Aug-2018.)
((∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ ∀𝑥𝐴 𝜑)
 
Theoremraaanlem 3374* Special case of raaan 3375 where 𝐴 is inhabited. (Contributed by Jim Kingdon, 6-Aug-2018.)
𝑦𝜑    &   𝑥𝜓       (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓)))
 
Theoremraaan 3375* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
 
Theoremraaanv 3376* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))
 
Theoremsbss 3377* Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
 
Theoremsbcssg 3378 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵𝐶𝐴 / 𝑥𝐵𝐴 / 𝑥𝐶))
 
2.1.15  Conditional operator
 
Syntaxcif 3379 Extend class notation to include the conditional operator. See df-if 3380 for a description. (In older databases this was denoted "ded".)
class if(𝜑, 𝐴, 𝐵)
 
Definitiondf-if 3380* Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if 𝜑 then 𝐴 else 𝐵." See iftrue 3384 and iffalse 3387 for its values. In mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise."

In the absence of excluded middle, this will tend to be useful where 𝜑 is decidable (in the sense of df-dc 779). (Contributed by NM, 15-May-1999.)

if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}
 
Theoremdfif6 3381* An alternate definition of the conditional operator df-if 3380 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})
 
Theoremifeq1 3382 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 = 𝐵 → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐶))
 
Theoremifeq2 3383 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
(𝐴 = 𝐵 → if(𝜑, 𝐶, 𝐴) = if(𝜑, 𝐶, 𝐵))
 
Theoremiftrue 3384 Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐴)
 
Theoremiftruei 3385 Inference associated with iftrue 3384. (Contributed by BJ, 7-Oct-2018.)
𝜑       if(𝜑, 𝐴, 𝐵) = 𝐴
 
Theoremiftrued 3386 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑𝜒)       (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐴)
 
Theoremiffalse 3387 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)
 
Theoremiffalsei 3388 Inference associated with iffalse 3387. (Contributed by BJ, 7-Oct-2018.)
¬ 𝜑       if(𝜑, 𝐴, 𝐵) = 𝐵
 
Theoremiffalsed 3389 Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
(𝜑 → ¬ 𝜒)       (𝜑 → if(𝜒, 𝐴, 𝐵) = 𝐵)
 
Theoremifnefalse 3390 When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3387 directly in this case. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)
 
Theoremifsbdc 3391 Distribute a function over an if-clause. (Contributed by Jim Kingdon, 1-Jan-2022.)
(if(𝜑, 𝐴, 𝐵) = 𝐴𝐶 = 𝐷)    &   (if(𝜑, 𝐴, 𝐵) = 𝐵𝐶 = 𝐸)       (DECID 𝜑𝐶 = if(𝜑, 𝐷, 𝐸))
 
Theoremdfif3 3392* Alternate definition of the conditional operator df-if 3380. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
𝐶 = {𝑥𝜑}       if(𝜑, 𝐴, 𝐵) = ((𝐴𝐶) ∪ (𝐵 ∩ (V ∖ 𝐶)))
 
Theoremifeq12 3393 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))
 
Theoremifeq1d 3394 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))
 
Theoremifeq2d 3395 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))
 
Theoremifeq12d 3396 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐷))
 
Theoremifbi 3397 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))
 
Theoremifbid 3398 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
(𝜑 → (𝜓𝜒))       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))
 
Theoremifbieq1d 3399 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜒, 𝐵, 𝐶))
 
Theoremifbieq2i 3400 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝜓)    &   𝐴 = 𝐵       if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)
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