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Theorem List for Metamath Proof Explorer - 4801-4900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremssexi 4801 The subset of a set is also a set. (Contributed by NM, 9-Sep-1993.)
𝐵 ∈ V    &   𝐴𝐵       𝐴 ∈ V

Theoremssexg 4802 The subset of a set is also a set. Exercise 3 of [TakeutiZaring] p. 22 (generalized). (Contributed by NM, 14-Aug-1994.)
((𝐴𝐵𝐵𝐶) → 𝐴 ∈ V)

Theoremssexd 4803 A subclass of a set is a set. Deduction form of ssexg 4802. (Contributed by David Moews, 1-May-2017.)
(𝜑𝐵𝐶)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ V)

Theoremprcssprc 4804 The superclass of a proper class is a proper class. (Contributed by AV, 27-Dec-2020.)
((𝐴𝐵𝐴 ∉ V) → 𝐵 ∉ V)

Theoremsselpwd 4805 Elementhood to a power set. (Contributed by Thierry Arnoux, 18-May-2020.)
(𝜑𝐵𝑉)    &   (𝜑𝐴𝐵)       (𝜑𝐴 ∈ 𝒫 𝐵)

Theoremdifexg 4806 Existence of a difference. (Contributed by NM, 26-May-1998.)
(𝐴𝑉 → (𝐴𝐵) ∈ V)

Theoremdifexi 4807 Existence of a difference, inference version of difexg 4806. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Revised by AV, 26-Mar-2021.)
𝐴 ∈ V       (𝐴𝐵) ∈ V

TheoremdifexOLD 4808 Obsolete version of difexi 4807 as of 26-Mar-2021. (Contributed by Glauco Siliprandi, 3-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐴𝑉       (𝐴𝐵) ∈ V

Theoremzfausab 4809* Separation Scheme (Aussonderung) in terms of a class abstraction. (Contributed by NM, 8-Jun-1994.)
𝐴 ∈ V       {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V

Theoremrabexg 4810* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 23-Oct-1999.)
(𝐴𝑉 → {𝑥𝐴𝜑} ∈ V)

Theoremrabex 4811* Separation Scheme in terms of a restricted class abstraction. (Contributed by NM, 19-Jul-1996.)
𝐴 ∈ V       {𝑥𝐴𝜑} ∈ V

Theoremrabexd 4812* Separation Scheme in terms of a restricted class abstraction, deduction form of rabex2 4813. (Contributed by AV, 16-Jul-2019.)
𝐵 = {𝑥𝐴𝜓}    &   (𝜑𝐴𝑉)       (𝜑𝐵 ∈ V)

Theoremrabex2 4813* Separation Scheme in terms of a restricted class abstraction. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
𝐵 = {𝑥𝐴𝜓}    &   𝐴 ∈ V       𝐵 ∈ V

Theoremrab2ex 4814* A class abstraction based on a class abstraction based on a set is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.)
𝐵 = {𝑦𝐴𝜓}    &   𝐴 ∈ V       {𝑥𝐵𝜑} ∈ V

Theoremrabex2OLD 4815* Obsolete version of rabex2 4813 as of 26-Mar-2021. (Contributed by AV, 16-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = {𝑥𝐴𝜓}    &   𝐴𝑉       𝐵 ∈ V

Theoremrab2exOLD 4816* Obsolete version of rab2ex 4814 as of 26-Mar-2021. (Contributed by AV, 16-Jul-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
𝐵 = {𝑦𝐴𝜓}    &   𝐴𝑉       {𝑥𝐵𝜑} ∈ V

Theoremelssabg 4817* Membership in a class abstraction involving a subset. Unlike elabg 3349, 𝐴 does not have to be a set. (Contributed by NM, 29-Aug-2006.)
(𝑥 = 𝐴 → (𝜑𝜓))       (𝐵𝑉 → (𝐴 ∈ {𝑥 ∣ (𝑥𝐵𝜑)} ↔ (𝐴𝐵𝜓)))

Theoremintex 4818 The intersection of a nonempty class exists. Exercise 5 of [TakeutiZaring] p. 44 and its converse. (Contributed by NM, 13-Aug-2002.)
(𝐴 ≠ ∅ ↔ 𝐴 ∈ V)

Theoremintnex 4819 If a class intersection is not a set, it must be the universe. (Contributed by NM, 3-Jul-2005.)
𝐴 ∈ V ↔ 𝐴 = V)

Theoremintexab 4820 The intersection of a nonempty class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝜑 {𝑥𝜑} ∈ V)

Theoremintexrab 4821 The intersection of a nonempty restricted class abstraction exists. (Contributed by NM, 21-Oct-2003.)
(∃𝑥𝐴 𝜑 {𝑥𝐴𝜑} ∈ V)

Theoremiinexg 4822* The existence of a class intersection. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by FL, 19-Sep-2011.)
((𝐴 ≠ ∅ ∧ ∀𝑥𝐴 𝐵𝐶) → 𝑥𝐴 𝐵 ∈ V)

Theoremintabs 4823* Absorption of a redundant conjunct in the intersection of a class abstraction. (Contributed by NM, 3-Jul-2005.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝑥 = {𝑦𝜓} → (𝜑𝜒))    &   ( {𝑦𝜓} ⊆ 𝐴𝜒)        {𝑥 ∣ (𝑥𝐴𝜑)} = {𝑥𝜑}

Theoreminuni 4824* The intersection of a union 𝐴 with a class 𝐵 is equal to the union of the intersections of each element of 𝐴 with 𝐵. (Contributed by FL, 24-Mar-2007.)
( 𝐴𝐵) = {𝑥 ∣ ∃𝑦𝐴 𝑥 = (𝑦𝐵)}

Theoremelpw2g 4825 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
(𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Theoremelpw2 4826 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 11-Oct-2007.)
𝐵 ∈ V       (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Theoremelpwi2 4827 Membership in a power class. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
𝐵𝑉    &   𝐴𝐵       𝐴 ∈ 𝒫 𝐵

Theorempwnss 4828 The power set of a set is never a subset. (Contributed by Stefan O'Rear, 22-Feb-2015.)
(𝐴𝑉 → ¬ 𝒫 𝐴𝐴)

Theorempwne 4829 No set equals its power set. The sethood antecedent is necessary; compare pwv 4431. (Contributed by NM, 17-Nov-2008.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
(𝐴𝑉 → 𝒫 𝐴𝐴)

2.2.5  Theorems requiring empty set existence

Theoremclass2set 4830* Construct, from any class 𝐴, a set equal to it when the class exists and equal to the empty set when the class is proper. This theorem shows that the constructed set always exists. (Contributed by NM, 16-Oct-2003.)
{𝑥𝐴𝐴 ∈ V} ∈ V

Theoremclass2seteq 4831* Equality theorem based on class2set 4830. (Contributed by NM, 13-Dec-2005.) (Proof shortened by Raph Levien, 30-Jun-2006.)
(𝐴𝑉 → {𝑥𝐴𝐴 ∈ V} = 𝐴)

Theorem0elpw 4832 Every power class contains the empty set. (Contributed by NM, 25-Oct-2007.)
∅ ∈ 𝒫 𝐴

Theorempwne0 4833 A power class is never empty. (Contributed by NM, 3-Sep-2018.)
𝒫 𝐴 ≠ ∅

Theorem0nep0 4834 The empty set and its power set are not equal. (Contributed by NM, 23-Dec-1993.)
∅ ≠ {∅}

Theorem0inp0 4835 Something cannot be equal to both the null set and the power set of the null set. (Contributed by NM, 21-Jun-1993.)
(𝐴 = ∅ → ¬ 𝐴 = {∅})

Theoremunidif0 4836 The removal of the empty set from a class does not affect its union. (Contributed by NM, 22-Mar-2004.)
(𝐴 ∖ {∅}) = 𝐴

Theoremiin0 4837* An indexed intersection of the empty set, with a nonempty index set, is empty. (Contributed by NM, 20-Oct-2005.)
(𝐴 ≠ ∅ ↔ 𝑥𝐴 ∅ = ∅)

Theoremnotzfaus 4838* In the Separation Scheme zfauscl 4781, we require that 𝑦 not occur in 𝜑 (which can be generalized to "not be free in"). Here we show special cases of 𝐴 and 𝜑 that result in a contradiction by violating this requirement. (Contributed by NM, 8-Feb-2006.)
𝐴 = {∅}    &   (𝜑 ↔ ¬ 𝑥𝑦)        ¬ ∃𝑦𝑥(𝑥𝑦 ↔ (𝑥𝐴𝜑))

Theoremintv 4839 The intersection of the universal class is empty. (Contributed by NM, 11-Sep-2008.)
V = ∅

Theoremaxpweq 4840* Two equivalent ways to express the Power Set Axiom. Note that ax-pow 4841 is not used by the proof. (Contributed by NM, 22-Jun-2009.)
𝐴 ∈ V       (𝒫 𝐴 ∈ V ↔ ∃𝑥𝑦(∀𝑧(𝑧𝑦𝑧𝐴) → 𝑦𝑥))

2.3  ZF Set Theory - add the Axiom of Power Sets

2.3.1  Introduce the Axiom of Power Sets

Axiomax-pow 4841* Axiom of Power Sets. An axiom of Zermelo-Fraenkel set theory. It states that a set 𝑦 exists that includes the power set of a given set 𝑥 i.e. contains every subset of 𝑥. The variant axpow2 4843 uses explicit subset notation. A version using class notation is pwex 4846. (Contributed by NM, 21-Jun-1993.)
𝑦𝑧(∀𝑤(𝑤𝑧𝑤𝑥) → 𝑧𝑦)

Theoremzfpow 4842* Axiom of Power Sets expressed with the fewest number of different variables. (Contributed by NM, 14-Aug-2003.)
𝑥𝑦(∀𝑥(𝑥𝑦𝑥𝑧) → 𝑦𝑥)

Theoremaxpow2 4843* A variant of the Axiom of Power Sets ax-pow 4841 using subset notation. Problem in [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)

Theoremaxpow3 4844* A variant of the Axiom of Power Sets ax-pow 4841. For any set 𝑥, there exists a set 𝑦 whose members are exactly the subsets of 𝑥 i.e. the power set of 𝑥. Axiom Pow of [BellMachover] p. 466. (Contributed by NM, 4-Jun-2006.)
𝑦𝑧(𝑧𝑥𝑧𝑦)

Theoremel 4845* Every set is an element of some other set. See elALT 4908 for a shorter proof using more axioms. (Contributed by NM, 4-Jan-2002.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝑦 𝑥𝑦

Theorempwex 4846 Power set axiom expressed in class notation. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
𝐴 ∈ V       𝒫 𝐴 ∈ V

Theoremvpwex 4847 The powerset of a setvar is a set. (Contributed by BJ, 3-May-2021.)
𝒫 𝑥 ∈ V

Theorempwexg 4848 Power set axiom expressed in class notation, with the sethood requirement as an antecedent. Axiom 4 of [TakeutiZaring] p. 17. (Contributed by NM, 30-Oct-2003.)
(𝐴𝑉 → 𝒫 𝐴 ∈ V)

Theoremabssexg 4849* Existence of a class of subsets. (Contributed by NM, 15-Jul-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)
(𝐴𝑉 → {𝑥 ∣ (𝑥𝐴𝜑)} ∈ V)

TheoremsnexALT 4850 Alternate proof of snex 4906 using Power Set (ax-pow 4841) instead of Pairing (ax-pr 4904). Unlike in the proof of zfpair 4902, Replacement (ax-rep 4769) is not needed. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
{𝐴} ∈ V

Theoremp0ex 4851 The power set of the empty set (the ordinal 1) is a set. See also p0exALT 4852. (Contributed by NM, 23-Dec-1993.)
{∅} ∈ V

Theoremp0exALT 4852 Alternate proof of p0ex 4851 which is quite different and longer if snexALT 4850 is expanded. (Contributed by NM, 23-Dec-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
{∅} ∈ V

Theorempp0ex 4853 The power set of the power set of the empty set (the ordinal 2) is a set. (Contributed by NM, 24-Jun-1993.)
{∅, {∅}} ∈ V

Theoremord3ex 4854 The ordinal number 3 is a set, proved without the Axiom of Union ax-un 6946. (Contributed by NM, 2-May-2009.)
{∅, {∅}, {∅, {∅}}} ∈ V

Theoremdtru 4855* At least two sets exist (or in terms of first-order logic, the universe of discourse has two or more objects). Note that we may not substitute the same variable for both 𝑥 and 𝑦 (as indicated by the distinct variable requirement), for otherwise we would contradict stdpc6 1956.

This theorem is proved directly from set theory axioms (no set theory definitions) and does not use ax-ext 2601 or ax-sep 4779. See dtruALT 4897 for a shorter proof using these axioms.

The proof makes use of dummy variables 𝑧 and 𝑤 which do not appear in the final theorem. They must be distinct from each other and from 𝑥 and 𝑦. In other words, if we were to substitute 𝑥 for 𝑧 throughout the proof, the proof would fail. (Contributed by NM, 7-Nov-2006.)

¬ ∀𝑥 𝑥 = 𝑦

Theoremaxc16b 4856* This theorem shows that axiom ax-c16 34003 is redundant in the presence of theorem dtru 4855, which states simply that at least two things exist. This justifies the remark at mmzfcnd.html#twoness (which links to this theorem). (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by NM, 7-Nov-2006.)
(∀𝑥 𝑥 = 𝑦 → (𝜑 → ∀𝑥𝜑))

Theoremeunex 4857 Existential uniqueness implies there is a value for which the wff argument is false. (Contributed by NM, 24-Oct-2010.)
(∃!𝑥𝜑 → ∃𝑥 ¬ 𝜑)

Theoremeusv1 4858* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.)
(∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃𝑦𝑥 𝑦 = 𝐴)

Theoremeusvnf 4859* Even if 𝑥 is free in 𝐴, it is effectively bound when 𝐴(𝑥) is single-valued. (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 14-Oct-2016.)
(∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)

Theoremeusvnfb 4860* Two ways to say that 𝐴(𝑥) is a set expression that does not depend on 𝑥. (Contributed by Mario Carneiro, 18-Nov-2016.)
(∃!𝑦𝑥 𝑦 = 𝐴 ↔ (𝑥𝐴𝐴 ∈ V))

Theoremeusv2i 4861* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 14-Oct-2010.) (Revised by Mario Carneiro, 18-Nov-2016.)
(∃!𝑦𝑥 𝑦 = 𝐴 → ∃!𝑦𝑥 𝑦 = 𝐴)

Theoremeusv2nf 4862* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by Mario Carneiro, 18-Nov-2016.)
𝐴 ∈ V       (∃!𝑦𝑥 𝑦 = 𝐴𝑥𝐴)

Theoremeusv2 4863* Two ways to express single-valuedness of a class expression 𝐴(𝑥). (Contributed by NM, 15-Oct-2010.) (Proof shortened by Mario Carneiro, 18-Nov-2016.)
𝐴 ∈ V       (∃!𝑦𝑥 𝑦 = 𝐴 ↔ ∃!𝑦𝑥 𝑦 = 𝐴)

Theoremreusv1 4864* Two ways to express single-valuedness of a class expression 𝐶(𝑦). (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
(∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))

Theoremreusv1OLD 4865* Obsolete proof of reusv1 4864 as of 7-Aug-2021. (Contributed by NM, 16-Dec-2012.) (Proof shortened by Mario Carneiro, 18-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(∃𝑦𝐵 𝜑 → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))

Theoremreusv2lem1 4866* Lemma for reusv2 4872. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝑦𝐴 𝑥 = 𝐵))

Theoremreusv2lem2 4867* Lemma for reusv2 4872. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
(∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)

Theoremreusv2lem2OLD 4868* Obsolete proof of reusv2lem2 4867 as of 7-Aug-2021. (Contributed by NM, 27-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(∃!𝑥𝑦𝐴 𝑥 = 𝐵 → ∃!𝑥𝑦𝐴 𝑥 = 𝐵)

Theoremreusv2lem3 4869* Lemma for reusv2 4872. (Contributed by NM, 14-Dec-2012.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
(∀𝑦𝐴 𝐵 ∈ V → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵))

Theoremreusv2lem4 4870* Lemma for reusv2 4872. (Contributed by NM, 13-Dec-2012.)
(∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝑦𝐵 ((𝐶𝐴𝜑) → 𝑥 = 𝐶))

Theoremreusv2lem5 4871* Lemma for reusv2 4872. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
((∀𝑦𝐵 𝐶𝐴𝐵 ≠ ∅) → (∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶 ↔ ∃!𝑥𝐴𝑦𝐵 𝑥 = 𝐶))

Theoremreusv2 4872* Two ways to express single-valuedness of a class expression 𝐶(𝑦) that is constant for those 𝑦𝐵 such that 𝜑. The first antecedent ensures that the constant value belongs to the existential uniqueness domain 𝐴, and the second ensures that 𝐶(𝑦) is evaluated for at least one 𝑦. (Contributed by NM, 4-Jan-2013.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
((∀𝑦𝐵 (𝜑𝐶𝐴) ∧ ∃𝑦𝐵 𝜑) → (∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) ↔ ∃!𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))

Theoremreusv3i 4873* Two ways of expressing existential uniqueness via an indirect equality. (Contributed by NM, 23-Dec-2012.)
(𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦 = 𝑧𝐶 = 𝐷)       (∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶) → ∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷))

Theoremreusv3 4874* Two ways to express single-valuedness of a class expression 𝐶(𝑦). See reusv1 4864 for the connection to uniqueness. (Contributed by NM, 27-Dec-2012.)
(𝑦 = 𝑧 → (𝜑𝜓))    &   (𝑦 = 𝑧𝐶 = 𝐷)       (∃𝑦𝐵 (𝜑𝐶𝐴) → (∀𝑦𝐵𝑧𝐵 ((𝜑𝜓) → 𝐶 = 𝐷) ↔ ∃𝑥𝐴𝑦𝐵 (𝜑𝑥 = 𝐶)))

Theoremeusv4 4875* Two ways to express single-valuedness of a class expression 𝐵(𝑥). (Contributed by NM, 27-Oct-2010.)
𝐵 ∈ V       (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃!𝑥𝑦𝐴 𝑥 = 𝐵)

Theoremalxfr 4876* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 18-Feb-2007.)
(𝑥 = 𝐴 → (𝜑𝜓))       ((∀𝑦 𝐴𝐵 ∧ ∀𝑥𝑦 𝑥 = 𝐴) → (∀𝑥𝜑 ↔ ∀𝑦𝜓))

Theoremralxfrd 4877* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (Proof shortened by JJ, 7-Aug-2021.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))

TheoremralxfrdOLD 4878* Obsolete proof of ralxfrd 4877 as of 7-Aug-2021. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))

Theoremrexxfrd 4879* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))

Theoremralxfr2d 4880* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.)
((𝜑𝑦𝐶) → 𝐴𝑉)    &   (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))

Theoremrexxfr2d 4881* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
((𝜑𝑦𝐶) → 𝐴𝑉)    &   (𝜑 → (𝑥𝐵 ↔ ∃𝑦𝐶 𝑥 = 𝐴))    &   ((𝜑𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))

Theoremralxfrd2 4882* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of ralxfrd 4877. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∀𝑥𝐵 𝜓 ↔ ∀𝑦𝐶 𝜒))

Theoremrexxfrd2 4883* Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Variant of rexxfrd 4879. (Contributed by Alexander van der Vekens, 25-Apr-2018.)
((𝜑𝑦𝐶) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃𝑦𝐶 𝑥 = 𝐴)    &   ((𝜑𝑦𝐶𝑥 = 𝐴) → (𝜓𝜒))       (𝜑 → (∃𝑥𝐵 𝜓 ↔ ∃𝑦𝐶 𝜒))

Theoremralxfr 4884* Transfer universal quantification from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
(𝑦𝐶𝐴𝐵)    &   (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)

TheoremralxfrALT 4885* Alternate proof of ralxfr 4884 which does not use ralxfrd 4877. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑦𝐶𝐴𝐵)    &   (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∀𝑥𝐵 𝜑 ↔ ∀𝑦𝐶 𝜓)

Theoremrexxfr 4886* Transfer existence from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
(𝑦𝐶𝐴𝐵)    &   (𝑥𝐵 → ∃𝑦𝐶 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃𝑥𝐵 𝜑 ↔ ∃𝑦𝐶 𝜓)

Theoremrabxfrd 4887* Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜒. (Contributed by NM, 16-Jan-2012.)
𝑦𝐵    &   𝑦𝐶    &   ((𝜑𝑦𝐷) → 𝐴𝐷)    &   (𝑥 = 𝐴 → (𝜓𝜒))    &   (𝑦 = 𝐵𝐴 = 𝐶)       ((𝜑𝐵𝐷) → (𝐶 ∈ {𝑥𝐷𝜓} ↔ 𝐵 ∈ {𝑦𝐷𝜒}))

Theoremrabxfr 4888* Class builder membership after substituting an expression 𝐴 (containing 𝑦) for 𝑥 in the class expression 𝜑. (Contributed by NM, 10-Jun-2005.)
𝑦𝐵    &   𝑦𝐶    &   (𝑦𝐷𝐴𝐷)    &   (𝑥 = 𝐴 → (𝜑𝜓))    &   (𝑦 = 𝐵𝐴 = 𝐶)       (𝐵𝐷 → (𝐶 ∈ {𝑥𝐷𝜑} ↔ 𝐵 ∈ {𝑦𝐷𝜓}))

Theoremreuxfr2d 4889* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 16-Jan-2012.) (Revised by NM, 16-Jun-2017.)
((𝜑𝑦𝐵) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃*𝑦𝐵 𝑥 = 𝐴)       (𝜑 → (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜓) ↔ ∃!𝑦𝐵 𝜓))

Theoremreuxfr2 4890* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. (Contributed by NM, 14-Nov-2004.) (Revised by NM, 16-Jun-2017.)
(𝑦𝐵𝐴𝐵)    &   (𝑥𝐵 → ∃*𝑦𝐵 𝑥 = 𝐴)       (∃!𝑥𝐵𝑦𝐵 (𝑥 = 𝐴𝜑) ↔ ∃!𝑦𝐵 𝜑)

Theoremreuxfrd 4891* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhypd 4893 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)
((𝜑𝑦𝐵) → 𝐴𝐵)    &   ((𝜑𝑥𝐵) → ∃!𝑦𝐵 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐵 𝜓 ↔ ∃!𝑦𝐵 𝜒))

Theoremreuxfr 4892* Transfer existential uniqueness from a variable 𝑥 to another variable 𝑦 contained in expression 𝐴. Use reuhyp 4894 to eliminate the second hypothesis. (Contributed by NM, 14-Nov-2004.)
(𝑦𝐵𝐴𝐵)    &   (𝑥𝐵 → ∃!𝑦𝐵 𝑥 = 𝐴)    &   (𝑥 = 𝐴 → (𝜑𝜓))       (∃!𝑥𝐵 𝜑 ↔ ∃!𝑦𝐵 𝜓)

Theoremreuhypd 4893* A theorem useful for eliminating the restricted existential uniqueness hypotheses in riotaxfrd 6639. (Contributed by NM, 16-Jan-2012.)
((𝜑𝑥𝐶) → 𝐵𝐶)    &   ((𝜑𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       ((𝜑𝑥𝐶) → ∃!𝑦𝐶 𝑥 = 𝐴)

Theoremreuhyp 4894* A theorem useful for eliminating the restricted existential uniqueness hypotheses in reuxfr 4892. (Contributed by NM, 15-Nov-2004.)
(𝑥𝐶𝐵𝐶)    &   ((𝑥𝐶𝑦𝐶) → (𝑥 = 𝐴𝑦 = 𝐵))       (𝑥𝐶 → ∃!𝑦𝐶 𝑥 = 𝐴)

Theoremnfnid 4895 A setvar variable is not free from itself. The proof relies on dtru 4855, that is, it is not true in a one-element domain. (Contributed by Mario Carneiro, 8-Oct-2016.)
¬ 𝑥𝑥

Theoremnfcvb 4896 The "distinctor" expression ¬ ∀𝑥𝑥 = 𝑦, stating that 𝑥 and 𝑦 are not the same variable, can be written in terms of in the obvious way. This theorem is not true in a one-element domain, because then 𝑥𝑦 and 𝑥𝑥 = 𝑦 will both be true. (Contributed by Mario Carneiro, 8-Oct-2016.)
(𝑥𝑦 ↔ ¬ ∀𝑥 𝑥 = 𝑦)

TheoremdtruALT 4897* Alternate proof of dtru 4855 which requires more axioms but is shorter and may be easier to understand.

Assuming that ZF set theory is consistent, we cannot prove this theorem unless we specify that 𝑥 and 𝑦 be distinct. Specifically, theorem spcev 3298 requires that 𝑥 must not occur in the subexpression ¬ 𝑦 = {∅} in step 4 nor in the subexpression ¬ 𝑦 = ∅ in step 9. The proof verifier will require that 𝑥 and 𝑦 be in a distinct variable group to ensure this. You can check this by deleting the \$d statement in set.mm and rerunning the verifier, which will print a detailed explanation of the distinct variable violation. (Contributed by NM, 15-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)

¬ ∀𝑥 𝑥 = 𝑦

Theoremdtrucor 4898* Corollary of dtru 4855. This example illustrates the danger of blindly trusting the standard Deduction Theorem without accounting for free variables: the theorem form of this deduction is not valid, as shown by dtrucor2 4899. (Contributed by NM, 27-Jun-2002.)
𝑥 = 𝑦       𝑥𝑦

Theoremdtrucor2 4899 The theorem form of the deduction dtrucor 4898 leads to a contradiction, as mentioned in the "Wrong!" example at mmdeduction.html#bad. (Contributed by NM, 20-Oct-2007.)
(𝑥 = 𝑦𝑥𝑦)       (𝜑 ∧ ¬ 𝜑)

Theoremdvdemo1 4900* Demonstration of a theorem (scheme) that requires (meta)variables 𝑥 and 𝑦 to be distinct, but no others. It bundles the theorem schemes 𝑥(𝑥 = 𝑦𝑥𝑥) and 𝑥(𝑥 = 𝑦𝑦𝑥). Compare dvdemo2 4901. ("Bundles" is a term introduced by Raph Levien.) (Contributed by NM, 1-Dec-2006.)
𝑥(𝑥 = 𝑦𝑧𝑥)

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