P. 24 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 2301-2400   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	eleq2d 2301	Deduction from equality to equivalence of membership. (Contributed by NM, 27-Dec-1993.)
⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (𝐶 ∈ 𝐴 ↔ 𝐶 ∈ 𝐵))
Theorem	eleq12d 2302	Deduction from equality to equivalence of membership. (Contributed by NM, 31-May-1994.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐷)    ⇒   ⊢ (𝜑 → (𝐴 ∈ 𝐶 ↔ 𝐵 ∈ 𝐷))
Theorem	eleq1a 2303	A transitive-type law relating membership and equality. (Contributed by NM, 9-Apr-1994.)
⊢ (𝐴 ∈ 𝐵 → (𝐶 = 𝐴 → 𝐶 ∈ 𝐵))
Theorem	eqeltri 2304	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ 𝐴 ∈ 𝐶
Theorem	eqeltrri 2305	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 = 𝐵    &   ⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ 𝐵 ∈ 𝐶
Theorem	eleqtri 2306	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ 𝐴 ∈ 𝐶
Theorem	eleqtrri 2307	Substitution of equal classes into membership relation. (Contributed by NM, 5-Aug-1993.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ 𝐴 ∈ 𝐶
Theorem	eqeltrd 2308	Substitution of equal classes into membership relation, deduction form. (Contributed by Raph Levien, 10-Dec-2002.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrrd 2309	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝐶)
Theorem	eleqtrd 2310	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrrd 2311	Deduction that substitutes equal classes into membership. (Contributed by NM, 14-Dec-2004.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	3eltr3i 2312	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ 𝐶 ∈ 𝐷
Theorem	3eltr4i 2313	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ 𝐶 ∈ 𝐷
Theorem	3eltr3d 2314	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐶)    &   ⊢ (𝜑 → 𝐵 = 𝐷)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	3eltr4d 2315	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐶 = 𝐴)    &   ⊢ (𝜑 → 𝐷 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	3eltr3g 2316	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐴 = 𝐶    &   ⊢ 𝐵 = 𝐷    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	3eltr4g 2317	Substitution of equal classes into membership relation. (Contributed by Mario Carneiro, 6-Jan-2017.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐶 = 𝐴    &   ⊢ 𝐷 = 𝐵    ⇒   ⊢ (𝜑 → 𝐶 ∈ 𝐷)
Theorem	eqeltrid 2318	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrrid 2319	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐵 = 𝐴    &   ⊢ (𝜑 → 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrid 2320	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ (𝜑 → 𝐵 = 𝐶)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrrid 2321	B membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ 𝐴 ∈ 𝐵    &   ⊢ (𝜑 → 𝐶 = 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrdi 2322	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eqeltrrdi 2323	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐵 = 𝐴)    &   ⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrdi 2324	A membership and equality inference. (Contributed by NM, 4-Jan-2006.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐵 = 𝐶    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleqtrrdi 2325	A membership and equality inference. (Contributed by NM, 24-Apr-2005.)
⊢ (𝜑 → 𝐴 ∈ 𝐵)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝜑 → 𝐴 ∈ 𝐶)
Theorem	eleq2s 2326	Substitution of equal classes into a membership antecedent. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝐴 ∈ 𝐵 → 𝜑)    &   ⊢ 𝐶 = 𝐵    ⇒   ⊢ (𝐴 ∈ 𝐶 → 𝜑)
Theorem	eqneltrd 2327	If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)
Theorem	eqneltrrd 2328	If a class is not an element of another class, an equal class is also not an element. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → 𝐴 = 𝐵)    &   ⊢ (𝜑 → ¬ 𝐴 ∈ 𝐶)    ⇒   ⊢ (𝜑 → ¬ 𝐵 ∈ 𝐶)
Theorem	neleqtrd 2329	If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)
Theorem	neleqtrrd 2330	If a class is not an element of another class, it is also not an element of an equal class. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)
Theorem	cleqh 2331*	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqf 2399. (Contributed by NM, 5-Aug-1993.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    &   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	nelneq 2332	A way of showing two classes are not equal. (Contributed by NM, 1-Apr-1997.)
⊢ ((𝐴 ∈ 𝐶 ∧ ¬ 𝐵 ∈ 𝐶) → ¬ 𝐴 = 𝐵)
Theorem	nelneq2 2333	A way of showing two classes are not equal. (Contributed by NM, 12-Jan-2002.)
⊢ ((𝐴 ∈ 𝐵 ∧ ¬ 𝐴 ∈ 𝐶) → ¬ 𝐵 = 𝐶)
Theorem	eqsb1lem 2334*	Lemma for eqsb1 2335. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)
Theorem	eqsb1 2335*	Substitution for the left-hand side in an equality. Class version of equsb3 2004. (Contributed by Rodolfo Medina, 28-Apr-2010.)
⊢ ([𝑦 / 𝑥]𝑥 = 𝐴 ↔ 𝑦 = 𝐴)
Theorem	clelsb1 2336*	Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2209). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
Theorem	clelsb2 2337*	Substitution for the second argument of the membership predicate in an atomic formula (class version of elsb2 2210). (Contributed by Jim Kingdon, 22-Nov-2018.)
⊢ ([𝑦 / 𝑥]𝐴 ∈ 𝑥 ↔ 𝐴 ∈ 𝑦)
Theorem	hbxfreq 2338	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfrbi 1520 for equivalence version. (Contributed by NM, 21-Aug-2007.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝑦 ∈ 𝐵 → ∀𝑥 𝑦 ∈ 𝐵)    ⇒   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
Theorem	hblem 2339*	Change the free variable of a hypothesis builder. (Contributed by NM, 5-Aug-1993.) (Revised by Andrew Salmon, 11-Jul-2011.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ (𝑧 ∈ 𝐴 → ∀𝑥 𝑧 ∈ 𝐴)
Theorem	abeq2 2340*	Equality of a class variable and a class abstraction (also called a class builder). Theorem 5.1 of [Quine] p. 34. This theorem shows the relationship between expressions with class abstractions and expressions with class variables. Note that abbi 2345 and its relatives are among those useful for converting theorems with class variables to equivalent theorems with wff variables, by first substituting a class abstraction for each class variable. Class variables can always be eliminated from a theorem to result in an equivalent theorem with wff variables, and vice-versa. The idea is roughly as follows. To convert a theorem with a wff variable 𝜑 (that has a free variable 𝑥) to a theorem with a class variable 𝐴, we substitute 𝑥 ∈ 𝐴 for 𝜑 throughout and simplify, where 𝐴 is a new class variable not already in the wff. Conversely, to convert a theorem with a class variable 𝐴 to one with 𝜑, we substitute {𝑥 ∣ 𝜑} for 𝐴 throughout and simplify, where 𝑥 and 𝜑 are new set and wff variables not already in the wff. For more information on class variables, see Quine pp. 15-21 and/or Takeuti and Zaring pp. 10-13. (Contributed by NM, 5-Aug-1993.)
⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
Theorem	abeq1 2341*	Equality of a class variable and a class abstraction. (Contributed by NM, 20-Aug-1993.)
⊢ ({𝑥 ∣ 𝜑} = 𝐴 ↔ ∀𝑥(𝜑 ↔ 𝑥 ∈ 𝐴))
Theorem	abeq2i 2342	Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 3-Apr-1996.)
⊢ 𝐴 = {𝑥 ∣ 𝜑}    ⇒   ⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)
Theorem	abeq1i 2343	Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 31-Jul-1994.)
⊢ {𝑥 ∣ 𝜑} = 𝐴    ⇒   ⊢ (𝜑 ↔ 𝑥 ∈ 𝐴)
Theorem	abeq2d 2344	Equality of a class variable and a class abstraction (deduction). (Contributed by NM, 16-Nov-1995.)
⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})    ⇒   ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))
Theorem	abbi 2345	Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})
Theorem	abbi2i 2346*	Equality of a class variable and a class abstraction (inference form). (Contributed by NM, 5-Aug-1993.)
⊢ (𝑥 ∈ 𝐴 ↔ 𝜑)    ⇒   ⊢ 𝐴 = {𝑥 ∣ 𝜑}
Theorem	abbii 2347	Equivalent wff's yield equal class abstractions (inference form). (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓}
Theorem	abbid 2348	Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
Theorem	abbidv 2349*	Equivalent wff's yield equal class abstractions (deduction form). (Contributed by NM, 10-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = {𝑥 ∣ 𝜒})
Theorem	abbi2dv 2350*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
Theorem	abbi1dv 2351*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝑥 ∈ 𝐴))    ⇒   ⊢ (𝜑 → {𝑥 ∣ 𝜓} = 𝐴)
Theorem	abid2 2352*	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 26-Dec-1993.)
⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴
Theorem	sb8ab 2353	Substitution of variable in class abstraction. (Contributed by Jim Kingdon, 27-Sep-2018.)
⊢ Ⅎ𝑦𝜑    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ [𝑦 / 𝑥]𝜑}
Theorem	cbvabw 2354*	Version of cbvab 2355 with a disjoint variable condition. (Contributed by GG, 10-Jan-2024.) Reduce axiom usage. (Revised by GG, 25-Aug-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
Theorem	cbvab 2355	Rule used to change bound variables, using implicit substitution. (Contributed by Andrew Salmon, 11-Jul-2011.)
⊢ Ⅎ𝑦𝜑    &   ⊢ Ⅎ𝑥𝜓    &   ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
Theorem	cbvabv 2356*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-May-1999.)
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))    ⇒   ⊢ {𝑥 ∣ 𝜑} = {𝑦 ∣ 𝜓}
Theorem	clelab 2357*	Membership of a class variable in a class abstraction. (Contributed by NM, 23-Dec-1993.)
⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝜑))
Theorem	clabel 2358*	Membership of a class abstraction in another class. (Contributed by NM, 17-Jan-2006.)
⊢ ({𝑥 ∣ 𝜑} ∈ 𝐴 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∀𝑥(𝑥 ∈ 𝑦 ↔ 𝜑)))
Theorem	sbab 2359*	The right-hand side of the second equality is a way of representing proper substitution of 𝑦 for 𝑥 into a class variable. (Contributed by NM, 14-Sep-2003.)
⊢ (𝑥 = 𝑦 → 𝐴 = {𝑧 ∣ [𝑦 / 𝑥]𝑧 ∈ 𝐴})
2.1.2.1  Elementary properties of class abstractions
Theorem	eqabdv 2360*	Deduction from a wff to a class abstraction. (Contributed by NM, 9-Jul-1994.) (Revised by Wolf Lammen, 6-May-2023.)
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝐴 = {𝑥 ∣ 𝜓})
2.1.3  Class form not-free predicate
Syntax	wnfc 2361	Extend wff definition to include the not-free predicate for classes.
wff Ⅎ𝑥𝐴
Theorem	nfcjust 2362*	Justification theorem for df-nfc 2363. (Contributed by Mario Carneiro, 13-Oct-2016.)
⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 ↔ ∀𝑧Ⅎ𝑥 𝑧 ∈ 𝐴)
Definition	df-nfc 2363*	Define the not-free predicate for classes. This is read "𝑥 is not free in 𝐴". Not-free means that the value of 𝑥 cannot affect the value of 𝐴, e.g., any occurrence of 𝑥 in 𝐴 is effectively bound by a quantifier or something that expands to one (such as "there exists at most one"). It is defined in terms of the not-free predicate df-nf 1509 for wffs; see that definition for more information. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfci 2364*	Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥 𝑦 ∈ 𝐴    ⇒   ⊢ Ⅎ𝑥𝐴
Theorem	nfcii 2365*	Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ Ⅎ𝑥𝐴
Theorem	nfcr 2366*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfcrii 2367*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
Theorem	nfcri 2368*	Consequence of the not-free predicate. (Note that unlike nfcr 2366, this does not require 𝑦 and 𝐴 to be disjoint.) (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
Theorem	nfcd 2369*	Deduce that a class 𝐴 does not have 𝑥 free in it. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐴)
Theorem	nfceqi 2370	Equality theorem for class not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝐴 = 𝐵    ⇒   ⊢ (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵)
Theorem	nfcxfr 2371	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝐴 = 𝐵    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥𝐴
Theorem	nfcxfrd 2372	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ 𝐴 = 𝐵    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝐴)
Theorem	nfceqdf 2373	An equality theorem for effectively not free. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ (𝜑 → 𝐴 = 𝐵)    ⇒   ⊢ (𝜑 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑥𝐵))
Theorem	nfcv 2374*	If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴
Theorem	nfcvd 2375*	If 𝑥 is disjoint from 𝐴, then 𝑥 is not free in 𝐴. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)
Theorem	nfab1 2376	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
Theorem	nfnfc1 2377	𝑥 is bound in Ⅎ𝑥𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥Ⅎ𝑥𝐴
Theorem	clelsb1f 2378	Substitution for the first argument of the membership predicate in an atomic formula (class version of elsb1 2209). (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.) (Revised by Thierry Arnoux, 13-Mar-2017.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
Theorem	nfab 2379	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥{𝑦 ∣ 𝜑}
Theorem	nfaba1 2380	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.)
⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}
Theorem	nfnfc 2381	Hypothesis builder for Ⅎ𝑦𝐴. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥Ⅎ𝑦𝐴
Theorem	nfeq 2382	Hypothesis builder for equality. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 = 𝐵
Theorem	nfel 2383	Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
Theorem	nfeq1 2384*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐴 = 𝐵
Theorem	nfel1 2385*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
Theorem	nfeq2 2386*	Hypothesis builder for equality, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 = 𝐵
Theorem	nfel2 2387*	Hypothesis builder for elementhood, special case. (Contributed by Mario Carneiro, 10-Oct-2016.)
⊢ Ⅎ𝑥𝐵    ⇒   ⊢ Ⅎ𝑥 𝐴 ∈ 𝐵
Theorem	nfcrd 2388*	Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝑦 ∈ 𝐴)
Theorem	nfeqd 2389	Hypothesis builder for equality. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 = 𝐵)
Theorem	nfeld 2390	Hypothesis builder for elementhood. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑥𝐵)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 𝐴 ∈ 𝐵)
Theorem	drnfc1 2391	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑥𝐴 ↔ Ⅎ𝑦𝐵))
Theorem	drnfc2 2392	Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ (∀𝑥 𝑥 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (∀𝑥 𝑥 = 𝑦 → (Ⅎ𝑧𝐴 ↔ Ⅎ𝑧𝐵))
Theorem	nfabdw 2393*	Bound-variable hypothesis builder for a class abstraction. Version of nfabd 2394 with a disjoint variable condition. (Contributed by Mario Carneiro, 8-Oct-2016.) (Revised by GG, 10-Jan-2024.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
Theorem	nfabd 2394	Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ Ⅎ𝑦𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})
Theorem	dvelimdc 2395	Deduction form of dvelimc 2396. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑧𝜑    &   ⊢ (𝜑 → Ⅎ𝑥𝐴)    &   ⊢ (𝜑 → Ⅎ𝑧𝐵)    &   ⊢ (𝜑 → (𝑧 = 𝑦 → 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵))
Theorem	dvelimc 2396	Version of dvelim 2070 for classes. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑧𝐵    &   ⊢ (𝑧 = 𝑦 → 𝐴 = 𝐵)    ⇒   ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝐵)
Theorem	nfcvf 2397	If 𝑥 and 𝑦 are distinct, then 𝑥 is not free in 𝑦. (Contributed by Mario Carneiro, 8-Oct-2016.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑥𝑦)
Theorem	nfcvf2 2398	If 𝑥 and 𝑦 are distinct, then 𝑦 is not free in 𝑥. (Contributed by Mario Carneiro, 5-Dec-2016.)
⊢ (¬ ∀𝑥 𝑥 = 𝑦 → Ⅎ𝑦𝑥)
Theorem	cleqf 2399	Establish equality between classes, using bound-variable hypotheses instead of distinct variable conditions. See also cleqh 2331. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    &   ⊢ Ⅎ𝑥𝐵    ⇒   ⊢ (𝐴 = 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵))
Theorem	abid2f 2400	A simplification of class abstraction. Theorem 5.2 of [Quine] p. 35. (Contributed by NM, 5-Sep-2011.) (Revised by Mario Carneiro, 7-Oct-2016.)
⊢ Ⅎ𝑥𝐴    ⇒   ⊢ {𝑥 ∣ 𝑥 ∈ 𝐴} = 𝐴