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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoreminundif 3901 The intersection and class difference of a class with another class unite to give the original class. (Contributed by Paul Chapman, 5-Jun-2009.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
((𝐴𝐵) ∪ (𝐴𝐵)) = 𝐴

Theoremdisjdif2 3902 The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)
((𝐴𝐵) = ∅ → (𝐴𝐵) = 𝐴)

Theoremdifun2 3903 Absorption of union by difference. Theorem 36 of [Suppes] p. 29. (Contributed by NM, 19-May-1998.)
((𝐴𝐵) ∖ 𝐵) = (𝐴𝐵)

Theoremundif 3904 Union of complementary parts into whole. (Contributed by NM, 22-Mar-1998.)
(𝐴𝐵 ↔ (𝐴 ∪ (𝐵𝐴)) = 𝐵)

Theoremssdifin0 3905 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 3906 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 = ∅)

Theoremssundif 3907 A condition equivalent to inclusion in the union of two classes. (Contributed by NM, 26-Mar-2007.)
(𝐴 ⊆ (𝐵𝐶) ↔ (𝐴𝐵) ⊆ 𝐶)

Theoremdifcom 3908 Swap the arguments of a class difference. (Contributed by NM, 29-Mar-2007.)
((𝐴𝐵) ⊆ 𝐶 ↔ (𝐴𝐶) ⊆ 𝐵)

Theorempssdifcom1 3909 Two ways to express overlapping subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
((𝐴𝐶𝐵𝐶) → ((𝐶𝐴) ⊊ 𝐵 ↔ (𝐶𝐵) ⊊ 𝐴))

Theorempssdifcom2 3910 Two ways to express non-covering pairs of subsets. (Contributed by Stefan O'Rear, 31-Oct-2014.)
((𝐴𝐶𝐵𝐶) → (𝐵 ⊊ (𝐶𝐴) ↔ 𝐴 ⊊ (𝐶𝐵)))

Theoremdifdifdir 3911 Distributive law for class difference. Exercise 4.8 of [Stoll] p. 16. (Contributed by NM, 18-Aug-2004.)
((𝐴𝐵) ∖ 𝐶) = ((𝐴𝐶) ∖ (𝐵𝐶))

Theoremuneqdifeq 3912 Two ways to say that 𝐴 and 𝐵 partition 𝐶 (when 𝐴 and 𝐵 don't overlap and 𝐴 is a part of 𝐶). (Contributed by FL, 17-Nov-2008.) (Proof shortened by JJ, 14-Jul-2021.)
((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))

TheoremuneqdifeqOLD 3913 Obsolete proof of uneqdifeq 3912 as of 14-Jul-2021. (Contributed by FL, 17-Nov-2008.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴𝐶 ∧ (𝐴𝐵) = ∅) → ((𝐴𝐵) = 𝐶 ↔ (𝐶𝐴) = 𝐵))

Theoremraldifeq 3914* Equality theorem for restricted universal quantifier. (Contributed by Thierry Arnoux, 6-Jul-2019.)
(𝜑𝐴𝐵)    &   (𝜑 → ∀𝑥 ∈ (𝐵𝐴)𝜓)       (𝜑 → (∀𝑥𝐴 𝜓 ↔ ∀𝑥𝐵 𝜓))

Theoremr19.2z 3915* Theorem 19.2 of [Margaris] p. 89 with restricted quantifiers (compare 19.2 1842). The restricted version is valid only when the domain of quantification is not empty. (Contributed by NM, 15-Nov-2003.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝜑) → ∃𝑥𝐴 𝜑)

Theoremr19.2zb 3916* A response to the notion that the condition 𝐴 ≠ ∅ can be removed in r19.2z 3915. Interestingly enough, 𝜑 does not figure in the left-hand side. (Contributed by Jeff Hankins, 24-Aug-2009.)
(𝐴 ≠ ∅ ↔ (∀𝑥𝐴 𝜑 → ∃𝑥𝐴 𝜑))

Theoremr19.3rz 3917* Restricted quantification of wff not containing quantified variable. (Contributed by FL, 3-Jan-2008.)
𝑥𝜑       (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Theoremr19.28z 3918* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
𝑥𝜑       (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))

Theoremr19.3rzv 3919* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 10-Mar-1997.)
(𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥𝐴 𝜑))

Theoremr19.9rzv 3920* Restricted quantification of wff not containing quantified variable. (Contributed by NM, 27-May-1998.)
(𝐴 ≠ ∅ → (𝜑 ↔ ∃𝑥𝐴 𝜑))

Theoremr19.28zv 3921* Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.)
(𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝐴 𝜓)))

Theoremr19.37zv 3922* Restricted quantifier version of Theorem 19.37 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Paul Chapman, 8-Oct-2007.)
(𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 → ∃𝑥𝐴 𝜓)))

Theoremr19.45zv 3923* Restricted version of Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝐴 𝜓)))

Theoremr19.44zv 3924* Restricted version of Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 27-May-1998.)
(𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (∃𝑥𝐴 𝜑𝜓)))

Theoremr19.27z 3925* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 26-Oct-2010.)
𝑥𝜓       (𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))

Theoremr19.27zv 3926* Restricted quantifier version of Theorem 19.27 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 19-Aug-2004.)
(𝐴 ≠ ∅ → (∀𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))

Theoremr19.36zv 3927* Restricted quantifier version of Theorem 19.36 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by NM, 20-Sep-2003.)
(𝐴 ≠ ∅ → (∃𝑥𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑𝜓)))

Theoremrzal 3928* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(𝐴 = ∅ → ∀𝑥𝐴 𝜑)

Theoremrexn0 3929* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
(∃𝑥𝐴 𝜑𝐴 ≠ ∅)

Theoremralidm 3930* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
(∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)

Theoremral0 3931 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
𝑥 ∈ ∅ 𝜑

TheoremrgenzOLD 3932* Obsolete as of 22-Jul-2021. (Contributed by NM, 8-Dec-2012.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝐴 ≠ ∅ ∧ 𝑥𝐴) → 𝜑)       𝑥𝐴 𝜑

Theoremralf0 3933* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.) (Proof shortened by JJ, 14-Jul-2021.)
¬ 𝜑       (∀𝑥𝐴 𝜑𝐴 = ∅)

Theoremralf0OLD 3934* Obsolete proof of ralf0 3933 as of 14-Jul-2021. (Contributed by NM, 26-Nov-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
¬ 𝜑       (∀𝑥𝐴 𝜑𝐴 = ∅)

Theoremraaan 3935* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
𝑦𝜑    &   𝑥𝜓       (∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))

Theoremraaanv 3936* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
(∀𝑥𝐴𝑦𝐴 (𝜑𝜓) ↔ (∀𝑥𝐴 𝜑 ∧ ∀𝑦𝐴 𝜓))

Theoremsbss 3937* 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 3938 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  "Weak deduction theorem" for set theory

In a Hilbert system of logic (which consists of a set of axioms, modus ponens, and the generalization rule), converting a deduction to a proof using the Deduction Theorem (taught in introductory logic books) involves an exponential increase of the number of steps as hypotheses are successively eliminated. Here is a trick that is not as general as the Deduction Theorem but requires only a linear increase in the number of steps.

The general problem: We want to convert a deduction P |- Q into a proof of the theorem |- P -> Q i.e., we want to eliminate the hypothesis P. Normally this is done using the Deduction (meta)Theorem, which looks at the microscopic steps of the deduction and usually doubles or triples the number of these microscopic steps for each hypothesis that is eliminated. We will look at a special case of this problem, without appealing to the Deduction Theorem.

We assume ZF with class notation. A and B are arbitrary (possibly proper) classes. P, Q, R, S and T are wffs.

We define the conditional operator, if(P, A, B), as follows: if(P, A, B) =def= { x | (x \in A & P) v (x \in B & -. P) } (where x does not occur in A, B, or P).

Lemma 1. A = if(P, A, B) -> (P <-> R), B = if(P, A, B) -> (S <-> R), S |- R Proof: Logic and Axiom of Extensionality.

Lemma 2. A = if(P, A, B) -> (Q <-> T), T |- P -> Q Proof: Logic and Axiom of Extensionality.

Here's a simple example that illustrates how it works. Suppose we have a deduction Ord A |- Tr A which means, "Assume A is an ordinal class. Then A is a transitive class." Note that A is a class variable that may be substituted with any class expression, so this is really a deduction scheme.

We want to convert this to a proof of the theorem (scheme) |- Ord A -> Tr A.

The catch is that we must be able to prove "Ord A" for at least one object A (and this is what makes it weaker than the ordinary Deduction Theorem). However, it is easy to prove |- Ord 0 (the empty set is ordinal). (For a typical textbook "theorem," i.e. deduction, there is usually at least one object satisfying each hypothesis, otherwise the theorem would not be very useful. We can always go back to the standard Deduction Theorem for those hypotheses where this is not the case.) Continuing with the example:

Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Ord A <-> Ord if(Ord A, A, 0)) (1) |- 0 = if(Ord A, A, 0) -> (Ord 0 <-> Ord if(Ord A, A, 0)) (2) From (1), (2) and |- Ord 0, Lemma 1 yields |- Ord if(Ord A, A, 0) (3) From (3) and substituting if(Ord A, A, 0) for A in the original deduction, |- Tr if(Ord A, A, 0) (4) Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Tr A <-> Tr if(Ord A, A, 0)) (5) From (4) and (5), Lemma 2 yields |- Ord A -> Tr A (Q.E.D.)

Syntaxcif 3939 Extend class notation to include the conditional operator. See df-if 3940 for a description. (In older databases this was denoted "ded".)
class if(𝜑, 𝐴, 𝐵)

Definitiondf-if 3940* Define the conditional operator. Read if(𝜑, 𝐴, 𝐵) as "if 𝜑 then 𝐴 else 𝐵." See iftrue 3945 and iffalse 3948 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 older versions of this database, this operator was denoted "ded" and called the "deduction class.")

An important use for us is in conjunction with the weak deduction theorem, which converts a hypothesis into an antecedent. In that role, 𝐴 is a class variable in the hypothesis and 𝐵 is a class (usually a constant) that makes the hypothesis true when it is substituted for 𝐴. See dedth 3992 for the main part of the weak deduction theorem, elimhyp 3999 to eliminate a hypothesis, and keephyp 4005 to keep a hypothesis. See the Deduction Theorem link on the Metamath Proof Explorer Home Page for a description of the weak deduction theorem. (Contributed by NM, 15-May-1999.)

if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐴𝜑) ∨ (𝑥𝐵 ∧ ¬ 𝜑))}

Theoremdfif2 3941* An alternate definition of the conditional operator df-if 3940 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
if(𝜑, 𝐴, 𝐵) = {𝑥 ∣ ((𝑥𝐵𝜑) → (𝑥𝐴𝜑))}

Theoremdfif6 3942* An alternate definition of the conditional operator df-if 3940 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
if(𝜑, 𝐴, 𝐵) = ({𝑥𝐴𝜑} ∪ {𝑥𝐵 ∣ ¬ 𝜑})

Theoremifeq1 3943 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 3944 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 3945 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 3946 Inference associated with iftrue 3945. (Contributed by BJ, 7-Oct-2018.)
𝜑       if(𝜑, 𝐴, 𝐵) = 𝐴

Theoremiftrued 3947 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 3948 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
𝜑 → if(𝜑, 𝐴, 𝐵) = 𝐵)

Theoremiffalsei 3949 Inference associated with iffalse 3948. (Contributed by BJ, 7-Oct-2018.)
¬ 𝜑       if(𝜑, 𝐴, 𝐵) = 𝐵

Theoremiffalsed 3950 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 3951 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 3948 directly in this case. It happens, e.g., in oevn0 7358. (Contributed by David A. Wheeler, 15-May-2015.)
(𝐴𝐵 → if(𝐴 = 𝐵, 𝐶, 𝐷) = 𝐷)

Theoremifsb 3952 Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
(if(𝜑, 𝐴, 𝐵) = 𝐴𝐶 = 𝐷)    &   (if(𝜑, 𝐴, 𝐵) = 𝐵𝐶 = 𝐸)       𝐶 = if(𝜑, 𝐷, 𝐸)

Theoremdfif3 3953* Alternate definition of the conditional operator df-if 3940. 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)))

Theoremdfif4 3954* Alternate definition of the conditional operator df-if 3940. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
𝐶 = {𝑥𝜑}       if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∩ ((𝐴 ∪ (V ∖ 𝐶)) ∩ (𝐵𝐶)))

Theoremdfif5 3955* Alternate definition of the conditional operator df-if 3940. Note that 𝜑 is independent of 𝑥 i.e. a constant true or false (see also ab0orv 3810). (Contributed by Gérard Lang, 18-Aug-2013.)
𝐶 = {𝑥𝜑}       if(𝜑, 𝐴, 𝐵) = ((𝐴𝐵) ∪ (((𝐴𝐵) ∩ 𝐶) ∪ ((𝐵𝐴) ∩ (V ∖ 𝐶))))

Theoremifeq12 3956 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
((𝐴 = 𝐵𝐶 = 𝐷) → if(𝜑, 𝐴, 𝐶) = if(𝜑, 𝐵, 𝐷))

Theoremifeq1d 3957 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Theoremifeq2d 3958 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
(𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Theoremifeq12d 3959 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 3960 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
((𝜑𝜓) → if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐴, 𝐵))

Theoremifbid 3961 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
(𝜑 → (𝜓𝜒))       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐴, 𝐵))

Theoremifbieq1d 3962 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 3963 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑𝜓)    &   𝐴 = 𝐵       if(𝜑, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵)

Theoremifbieq2d 3964 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜒, 𝐶, 𝐵))

Theoremifbieq12i 3965 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
(𝜑𝜓)    &   𝐴 = 𝐶    &   𝐵 = 𝐷       if(𝜑, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷)

Theoremifbieq12d 3966 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
(𝜑 → (𝜓𝜒))    &   (𝜑𝐴 = 𝐶)    &   (𝜑𝐵 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Theoremnfifd 3967 Deduction version of nfif 3968. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑𝑥𝐴)    &   (𝜑𝑥𝐵)       (𝜑𝑥if(𝜓, 𝐴, 𝐵))

Theoremnfif 3968 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
𝑥𝜑    &   𝑥𝐴    &   𝑥𝐵       𝑥if(𝜑, 𝐴, 𝐵)

Theoremifeq1da 3969 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑𝜓) → 𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐴, 𝐶) = if(𝜓, 𝐵, 𝐶))

Theoremifeq2da 3970 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑 ∧ ¬ 𝜓) → 𝐴 = 𝐵)       (𝜑 → if(𝜓, 𝐶, 𝐴) = if(𝜓, 𝐶, 𝐵))

Theoremifeq12da 3971 Equivalence deduction for conditional operators. (Contributed by Wolf Lammen, 24-Jun-2021.)
((𝜑𝜓) → 𝐴 = 𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜓, 𝐶, 𝐷))

Theoremifbieq12d2 3972 Equivalence deduction for conditional operators. (Contributed by Thierry Arnoux, 14-Feb-2017.) (Proof shortened by Wolf Lammen, 24-Jun-2021.)
(𝜑 → (𝜓𝜒))    &   ((𝜑𝜓) → 𝐴 = 𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐷)       (𝜑 → if(𝜓, 𝐴, 𝐵) = if(𝜒, 𝐶, 𝐷))

Theoremifclda 3973 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
((𝜑𝜓) → 𝐴𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

Theoremifeqda 3974 Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
((𝜑𝜓) → 𝐴 = 𝐶)    &   ((𝜑 ∧ ¬ 𝜓) → 𝐵 = 𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) = 𝐶)

Theoremelimif 3975 Elimination of a conditional operator contained in a wff 𝜓. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)
(if(𝜑, 𝐴, 𝐵) = 𝐴 → (𝜓𝜒))    &   (if(𝜑, 𝐴, 𝐵) = 𝐵 → (𝜓𝜃))       (𝜓 ↔ ((𝜑𝜒) ∨ (¬ 𝜑𝜃)))

Theoremifbothda 3976 A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))    &   ((𝜂𝜑) → 𝜓)    &   ((𝜂 ∧ ¬ 𝜑) → 𝜒)       (𝜂𝜃)

Theoremifboth 3977 A wff 𝜃 containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜃))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜃))       ((𝜓𝜒) → 𝜃)

Theoremifid 3978 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
if(𝜑, 𝐴, 𝐴) = 𝐴

Theoremeqif 3979 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
(𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∨ (¬ 𝜑𝐴 = 𝐶)))

Theoremifval 3980 Another expression of the value of the if predicate, analogous to eqif 3979. See also the more specialized iftrue 3945 and iffalse 3948. (Contributed by BJ, 6-Apr-2019.)
(𝐴 = if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴 = 𝐵) ∧ (¬ 𝜑𝐴 = 𝐶)))

Theoremelif 3981 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
(𝐴 ∈ if(𝜑, 𝐵, 𝐶) ↔ ((𝜑𝐴𝐵) ∨ (¬ 𝜑𝐴𝐶)))

Theoremifel 3982 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
(if(𝜑, 𝐴, 𝐵) ∈ 𝐶 ↔ ((𝜑𝐴𝐶) ∨ (¬ 𝜑𝐵𝐶)))

Theoremifcl 3983 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
((𝐴𝐶𝐵𝐶) → if(𝜑, 𝐴, 𝐵) ∈ 𝐶)

Theoremifcld 3984 Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.)
(𝜑𝐴𝐶)    &   (𝜑𝐵𝐶)       (𝜑 → if(𝜓, 𝐴, 𝐵) ∈ 𝐶)

Theoremifeqor 3985 The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
(if(𝜑, 𝐴, 𝐵) = 𝐴 ∨ if(𝜑, 𝐴, 𝐵) = 𝐵)

Theoremifnot 3986 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
if(¬ 𝜑, 𝐴, 𝐵) = if(𝜑, 𝐵, 𝐴)

Theoremifan 3987 Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, if(𝜓, 𝐴, 𝐵), 𝐵)

Theoremifor 3988 Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
if((𝜑𝜓), 𝐴, 𝐵) = if(𝜑, 𝐴, if(𝜓, 𝐴, 𝐵))

Theorem2if2 3989 Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)
((𝜑𝜓) → 𝐷 = 𝐴)    &   ((𝜑 ∧ ¬ 𝜓𝜃) → 𝐷 = 𝐵)    &   ((𝜑 ∧ ¬ 𝜓 ∧ ¬ 𝜃) → 𝐷 = 𝐶)       (𝜑𝐷 = if(𝜓, 𝐴, if(𝜃, 𝐵, 𝐶)))

Theoremifcomnan 3990 Commute the conditions in two nested conditionals if both conditions are not simultaneously true. (Contributed by SO, 15-Jul-2018.)
(¬ (𝜑𝜓) → if(𝜑, 𝐴, if(𝜓, 𝐵, 𝐶)) = if(𝜓, 𝐵, if(𝜑, 𝐴, 𝐶)))

Theoremcsbif 3991 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)
𝐴 / 𝑥if(𝜑, 𝐵, 𝐶) = if([𝐴 / 𝑥]𝜑, 𝐴 / 𝑥𝐵, 𝐴 / 𝑥𝐶)

Theoremdedth 3992 Weak deduction theorem that eliminates a hypothesis 𝜑, making it become an antecedent. We assume that a proof exists for 𝜑 when the class variable 𝐴 is replaced with a specific class 𝐵. The hypothesis 𝜒 should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 3999. If the inference has other hypotheses with class variable 𝐴, these can be kept by assigning keephyp 4005 to them. For more information, see the Weak Deduction Theorem page mmdeduction.html. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜓𝜒))    &   𝜒       (𝜑𝜓)

Theoremdedth2h 3993 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3996 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3992. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜒𝜃))    &   (𝐵 = if(𝜓, 𝐵, 𝐷) → (𝜃𝜏))    &   𝜏       ((𝜑𝜓) → 𝜒)

Theoremdedth3h 3994 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3993. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜃𝜏))    &   (𝐵 = if(𝜓, 𝐵, 𝑅) → (𝜏𝜂))    &   (𝐶 = if(𝜒, 𝐶, 𝑆) → (𝜂𝜁))    &   𝜁       ((𝜑𝜓𝜒) → 𝜃)

Theoremdedth4h 3995 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3993. (Contributed by NM, 16-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜏𝜂))    &   (𝐵 = if(𝜓, 𝐵, 𝑆) → (𝜂𝜁))    &   (𝐶 = if(𝜒, 𝐶, 𝐹) → (𝜁𝜎))    &   (𝐷 = if(𝜃, 𝐷, 𝐺) → (𝜎𝜌))    &   𝜌       (((𝜑𝜓) ∧ (𝜒𝜃)) → 𝜏)

Theoremdedth2v 3996 Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 3993 is simpler to use. See also comments in dedth 3992. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   𝜃       (𝜑𝜓)

Theoremdedth3v 3997 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3996. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝐷) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑅) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑆) → (𝜃𝜏))    &   𝜏       (𝜑𝜓)

Theoremdedth4v 3998 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3996. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
(𝐴 = if(𝜑, 𝐴, 𝑅) → (𝜓𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝑆) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐶, 𝑇) → (𝜃𝜏))    &   (𝐷 = if(𝜑, 𝐷, 𝑈) → (𝜏𝜂))    &   𝜂       (𝜑𝜓)

Theoremelimhyp 3999 Eliminate a hypothesis containing class variable 𝐴 when it is known for a specific class 𝐵. For more information, see comments in dedth 3992. (Contributed by NM, 15-May-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐵) → (𝜑𝜓))    &   (𝐵 = if(𝜑, 𝐴, 𝐵) → (𝜒𝜓))    &   𝜒       𝜓

Theoremelimhyp2v 4000 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
(𝐴 = if(𝜑, 𝐴, 𝐶) → (𝜑𝜒))    &   (𝐵 = if(𝜑, 𝐵, 𝐷) → (𝜒𝜃))    &   (𝐶 = if(𝜑, 𝐴, 𝐶) → (𝜏𝜂))    &   (𝐷 = if(𝜑, 𝐵, 𝐷) → (𝜂𝜃))    &   𝜏       𝜃

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