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Theorem List for Metamath Proof Explorer - 36101-36200   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-line2 36101* Define the Line function. This function generates the line passing through the distinct points 𝑎 and 𝑏. Adapted from definition 6.14 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 25-Oct-2013.)
Line = {⟨⟨𝑎, 𝑏⟩, 𝑙⟩ ∣ ∃𝑛 ∈ ℕ ((𝑎 ∈ (𝔼‘𝑛) ∧ 𝑏 ∈ (𝔼‘𝑛) ∧ 𝑎𝑏) ∧ 𝑙 = [⟨𝑎, 𝑏⟩] Colinear )}
 
Definitiondf-ray 36102* Define the Ray function. This function generates the set of all points that lie on the ray starting at 𝑝 and passing through 𝑎. Definition 6.8 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 21-Oct-2013.)
Ray = {⟨⟨𝑝, 𝑎⟩, 𝑟⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ (𝔼‘𝑛) ∧ 𝑎 ∈ (𝔼‘𝑛) ∧ 𝑝𝑎) ∧ 𝑟 = {𝑥 ∈ (𝔼‘𝑛) ∣ 𝑝OutsideOf⟨𝑎, 𝑥⟩})}
 
Definitiondf-lines2 36103 Define the set of all lines. Definition 6.14, part 2 of [Schwabhauser] p. 45. See ellines 36116 for membership. (Contributed by Scott Fenton, 28-Oct-2013.)
LinesEE = ran Line
 
Theoremfunray 36104 Show that the Ray relationship is a function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Ray
 
Theoremfvray 36105* Calculate the value of the Ray function. (Contributed by Scott Fenton, 21-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁) ∧ 𝑃𝐴)) → (𝑃Ray𝐴) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑃OutsideOf⟨𝐴, 𝑥⟩})
 
Theoremfunline 36106 Show that the Line relationship is a function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Fun Line
 
Theoremlinedegen 36107 When Line is applied with the same argument, the result is the empty set. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴Line𝐴) = ∅
 
Theoremfvline 36108* Calculate the value of the Line function. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → (𝐴Line𝐵) = {𝑥𝑥 Colinear ⟨𝐴, 𝐵⟩})
 
Theoremliness 36109 A line is a subset of the space its two points lie in. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → (𝐴Line𝐵) ⊆ (𝔼‘𝑁))
 
Theoremfvline2 36110* Alternate definition of a line. (Contributed by Scott Fenton, 25-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐴𝐵)) → (𝐴Line𝐵) = {𝑥 ∈ (𝔼‘𝑁) ∣ 𝑥 Colinear ⟨𝐴, 𝐵⟩})
 
Theoremlineunray 36111 A line is composed of a point and the two rays emerging from it. Theorem 6.15 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 26-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑅 ∈ (𝔼‘𝑁)) ∧ (𝑃𝑄𝑃𝑅)) → (𝑃 Btwn ⟨𝑄, 𝑅⟩ → (𝑃Line𝑄) = (((𝑃Ray𝑄) ∪ {𝑃}) ∪ (𝑃Ray𝑅))))
 
Theoremlineelsb2 36112 If 𝑆 lies on 𝑃𝑄, then 𝑃𝑄 = 𝑃𝑆. Theorem 6.16 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄) ∧ (𝑆 ∈ (𝔼‘𝑁) ∧ 𝑃𝑆)) → (𝑆 ∈ (𝑃Line𝑄) → (𝑃Line𝑄) = (𝑃Line𝑆)))
 
Theoremlinerflx1 36113 Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → 𝑃 ∈ (𝑃Line𝑄))
 
Theoremlinecom 36114 Commutativity law for lines. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → (𝑃Line𝑄) = (𝑄Line𝑃))
 
Theoremlinerflx2 36115 Reflexivity law for line membership. Part of theorem 6.17 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → 𝑄 ∈ (𝑃Line𝑄))
 
Theoremellines 36116* Membership in the set of all lines. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
(𝐴 ∈ LinesEE ↔ ∃𝑛 ∈ ℕ ∃𝑝 ∈ (𝔼‘𝑛)∃𝑞 ∈ (𝔼‘𝑛)(𝑝𝑞𝐴 = (𝑝Line𝑞)))
 
Theoremlinethru 36117 If 𝐴 is a line containing two distinct points 𝑃 and 𝑄, then 𝐴 is the line through 𝑃 and 𝑄. Theorem 6.18 of [Schwabhauser] p. 45. (Contributed by Scott Fenton, 28-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴 ∈ LinesEE ∧ (𝑃𝐴𝑄𝐴) ∧ 𝑃𝑄) → 𝐴 = (𝑃Line𝑄))
 
Theoremhilbert1.1 36118* There is a line through any two distinct points. Hilbert's axiom I.1 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → ∃𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))
 
Theoremhilbert1.2 36119* There is at most one line through any two distinct points. Hilbert's axiom I.2 for geometry. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by NM, 17-Jun-2017.)
(𝑃𝑄 → ∃*𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))
 
Theoremlinethrueu 36120* There is a unique line going through any two distinct points. Theorem 6.19 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝑁 ∈ ℕ ∧ (𝑃 ∈ (𝔼‘𝑁) ∧ 𝑄 ∈ (𝔼‘𝑁) ∧ 𝑃𝑄)) → ∃!𝑥 ∈ LinesEE (𝑃𝑥𝑄𝑥))
 
Theoremlineintmo 36121* Two distinct lines intersect in at most one point. Theorem 6.21 of [Schwabhauser] p. 46. (Contributed by Scott Fenton, 29-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
((𝐴 ∈ LinesEE ∧ 𝐵 ∈ LinesEE ∧ 𝐴𝐵) → ∃*𝑥(𝑥𝐴𝑥𝐵))
 
21.11.19  Forward difference
 
Syntaxcfwddif 36122 Declare the syntax for the forward difference operator.
class
 
Definitiondf-fwddif 36123* Define the forward difference operator. This is a discrete analogue of the derivative operator. Definition 2.42 of [GramKnuthPat], p. 47. (Contributed by Scott Fenton, 18-May-2020.)
△ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓𝑥))))
 
Syntaxcfwddifn 36124 Declare the syntax for the nth forward difference operator.
class n
 
Definitiondf-fwddifn 36125* Define the nth forward difference operator. This works out to be the forward difference operator iterated 𝑛 times. (Contributed by Scott Fenton, 28-May-2020.)
n = (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))))
 
Theoremfwddifval 36126 Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋𝐴)    &   (𝜑 → (𝑋 + 1) ∈ 𝐴)       (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
 
Theoremfwddifnval 36127* The value of the forward difference operator at a point. (Contributed by Scott Fenton, 28-May-2020.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴)       (𝜑 → ((𝑁n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
 
Theoremfwddifn0 36128 The value of the n-iterated forward difference operator at zero is just the function value. (Contributed by Scott Fenton, 28-May-2020.)
(𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋𝐴)       (𝜑 → ((0 △n 𝐹)‘𝑋) = (𝐹𝑋))
 
Theoremfwddifnp1 36129* The value of the n-iterated forward difference at a successor. (Contributed by Scott Fenton, 28-May-2020.)
(𝜑𝑁 ∈ ℕ0)    &   (𝜑𝐴 ⊆ ℂ)    &   (𝜑𝐹:𝐴⟶ℂ)    &   (𝜑𝑋 ∈ ℂ)    &   ((𝜑𝑘 ∈ (0...(𝑁 + 1))) → (𝑋 + 𝑘) ∈ 𝐴)       (𝜑 → (((𝑁 + 1) △n 𝐹)‘𝑋) = (((𝑁n 𝐹)‘(𝑋 + 1)) − ((𝑁n 𝐹)‘𝑋)))
 
21.11.20  Rank theorems
 
Theoremrankung 36130 The rank of the union of two sets. Closed form of rankun 9925. (Contributed by Scott Fenton, 15-Jul-2015.)
((𝐴𝑉𝐵𝑊) → (rank‘(𝐴𝐵)) = ((rank‘𝐴) ∪ (rank‘𝐵)))
 
Theoremranksng 36131 The rank of a singleton. Closed form of ranksn 9923. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴𝑉 → (rank‘{𝐴}) = suc (rank‘𝐴))
 
Theoremrankelg 36132 The membership relation is inherited by the rank function. Closed form of rankel 9908. (Contributed by Scott Fenton, 16-Jul-2015.)
((𝐵𝑉𝐴𝐵) → (rank‘𝐴) ∈ (rank‘𝐵))
 
Theoremrankpwg 36133 The rank of a power set. Closed form of rankpw 9912. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴𝑉 → (rank‘𝒫 𝐴) = suc (rank‘𝐴))
 
Theoremrank0 36134 The rank of the empty set is . (Contributed by Scott Fenton, 17-Jul-2015.)
(rank‘∅) = ∅
 
Theoremrankeq1o 36135 The only set with rank 1o is the singleton of the empty set. (Contributed by Scott Fenton, 17-Jul-2015.)
((rank‘𝐴) = 1o𝐴 = {∅})
 
21.11.21  Hereditarily Finite Sets
 
Syntaxchf 36136 The constant Hf is a class.
class Hf
 
Definitiondf-hf 36137 Define the hereditarily finite sets. These are the finite sets whose elements are finite, and so forth. (Contributed by Scott Fenton, 9-Jul-2015.)
Hf = (𝑅1 “ ω)
 
Theoremelhf 36138* Membership in the hereditarily finite sets. (Contributed by Scott Fenton, 9-Jul-2015.)
(𝐴 ∈ Hf ↔ ∃𝑥 ∈ ω 𝐴 ∈ (𝑅1𝑥))
 
Theoremelhf2 36139 Alternate form of membership in the hereditarily finite sets. (Contributed by Scott Fenton, 13-Jul-2015.)
𝐴 ∈ V       (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω)
 
Theoremelhf2g 36140 Hereditarily finiteness via rank. Closed form of elhf2 36139. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴𝑉 → (𝐴 ∈ Hf ↔ (rank‘𝐴) ∈ ω))
 
Theorem0hf 36141 The empty set is a hereditarily finite set. (Contributed by Scott Fenton, 9-Jul-2015.)
∅ ∈ Hf
 
Theoremhfun 36142 The union of two HF sets is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴𝐵) ∈ Hf )
 
Theoremhfsn 36143 The singleton of an HF set is an HF set. (Contributed by Scott Fenton, 15-Jul-2015.)
(𝐴 ∈ Hf → {𝐴} ∈ Hf )
 
Theoremhfadj 36144 Adjoining one HF element to an HF set preserves HF status. (Contributed by Scott Fenton, 15-Jul-2015.)
((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 ∪ {𝐵}) ∈ Hf )
 
Theoremhfelhf 36145 Any member of an HF set is itself an HF set. (Contributed by Scott Fenton, 16-Jul-2015.)
((𝐴𝐵𝐵 ∈ Hf ) → 𝐴 ∈ Hf )
 
Theoremhftr 36146 The class of all hereditarily finite sets is transitive. (Contributed by Scott Fenton, 16-Jul-2015.)
Tr Hf
 
Theoremhfext 36147* Extensionality for HF sets depends only on comparison of HF elements. (Contributed by Scott Fenton, 16-Jul-2015.)
((𝐴 ∈ Hf ∧ 𝐵 ∈ Hf ) → (𝐴 = 𝐵 ↔ ∀𝑥 ∈ Hf (𝑥𝐴𝑥𝐵)))
 
Theoremhfuni 36148 The union of an HF set is itself hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴 ∈ Hf → 𝐴 ∈ Hf )
 
Theoremhfpw 36149 The power class of an HF set is hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
(𝐴 ∈ Hf → 𝒫 𝐴 ∈ Hf )
 
Theoremhfninf 36150 ω is not hereditarily finite. (Contributed by Scott Fenton, 16-Jul-2015.)
¬ ω ∈ Hf
 
21.12  Mathbox for Gino Giotto
 
21.12.1  Equality theorems.
 
21.12.1.1  Inference versions.
 
Theoremrmoeqi 36151 Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵       (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐵 𝜓)
 
Theoremrmoeqbii 36152 Equality inference for restricted at-most-one quantifier. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜓𝜒)       (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐵 𝜒)
 
Theoremreueqi 36153 Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵       (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐵 𝜓)
 
Theoremreueqbii 36154 Equality inference for restricted existential uniqueness quantifier. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜓𝜒)       (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐵 𝜒)
 
Theoremsbceqbii 36155 Formula-building inference for class substitution. General version of sbcbii 3865. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜑𝜓)       ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜓)
 
Theoremdisjeq1i 36156 Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵       (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐶)
 
Theoremdisjeq12i 36157 Equality theorem for disjoint collection. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷       (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷)
 
Theoremrabeqbii 36158 Equality theorem for restricted class abstractions. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜑𝜓)       {𝑥𝐴𝜑} = {𝑥𝐵𝜓}
 
Theoremiuneq12i 36159 Equality theorem for indexed union. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷        𝑥𝐴 𝐶 = 𝑥𝐵 𝐷
 
Theoremiineq1i 36160 Equality theorem for indexed intersection. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵        𝑥𝐴 𝐶 = 𝑥𝐵 𝐶
 
Theoremiineq12i 36161 Equality theorem for indexed intersection. Inference version. General version of iineq1i 36160. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷        𝑥𝐴 𝐶 = 𝑥𝐵 𝐷
 
Theoremriotaeqbii 36162 Equivalent wff's and equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝜑𝜓)       (𝑥𝐴 𝜑) = (𝑥𝐵 𝜓)
 
Theoremriotaeqi 36163 Equal domains yield equal restricted iotas. Inference version. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵       (𝑥𝐴 𝜑) = (𝑥𝐵 𝜑)
 
Theoremixpeq1i 36164 Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵       X𝑥𝐴 𝐶 = X𝑥𝐵 𝐶
 
Theoremixpeq12i 36165 Equality inference for infinite Cartesian product. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷       X𝑥𝐴 𝐶 = X𝑥𝐵 𝐷
 
Theoremsumeq2si 36166 Equality inference for sum. (Contributed by GG, 1-Sep-2025.)
𝐵 = 𝐶       Σ𝑘𝐴 𝐵 = Σ𝑘𝐴 𝐶
 
Theoremsumeq12si 36167 Equality inference for sum. General version of sumeq2si 36166. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷       Σ𝑥𝐴 𝐶 = Σ𝑥𝐵 𝐷
 
Theoremprodeq2si 36168 Equality inference for product. (Contributed by GG, 1-Sep-2025.)
𝐵 = 𝐶       𝑘𝐴 𝐵 = ∏𝑘𝐴 𝐶
 
Theoremprodeq12si 36169 Equality inference for product. General version of prodeq2si 36168. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝑥𝐴 𝐶 = ∏𝑥𝐵 𝐷
 
Theoremitgeq12i 36170 Equality inference for an integral. General version of itgeq1i 36171 and itgeq2i 36172. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷       𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥
 
Theoremitgeq1i 36171 Equality inference for an integral. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵       𝐴𝐶 d𝑥 = ∫𝐵𝐶 d𝑥
 
Theoremitgeq2i 36172 Equality inference for an integral. (Contributed by GG, 1-Sep-2025.)
𝐵 = 𝐶       𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥
 
Theoremditgeq123i 36173 Equality inference for the directed integral. General version of ditgeq12i 36174 and ditgeq3i 36175. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷    &   𝐸 = 𝐹       ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]𝐹 d𝑥
 
Theoremditgeq12i 36174 Equality inference for the directed integral. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   𝐶 = 𝐷       ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]𝐸 d𝑥
 
Theoremditgeq3i 36175 Equality inference for the directed integral. (Contributed by GG, 1-Sep-2025.)
𝐶 = 𝐷       ⨜[𝐴𝐵]𝐶 d𝑥 = ⨜[𝐴𝐵]𝐷 d𝑥
 
21.12.1.2  Deduction versions.
 
Theoremrmoeqdv 36176* Formula-building rule for restricted at-most-one quantifier. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐵 𝜓))
 
Theoremrmoeqbidv 36177* Formula-building rule for restricted at-most-one quantifier. Deduction form. General version of rmobidv 3405. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃*𝑥𝐴 𝜓 ↔ ∃*𝑥𝐵 𝜒))
 
Theoremreueqdv 36178* Formula-building rule for restricted existential uniqueness quantifier. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐵 𝜓))
 
Theoremreueqbidv 36179* Formula-building rule for restricted existential uniqueness quantifier. Deduction form. General version of reubidv 3406. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃!𝑥𝐴 𝜓 ↔ ∃!𝑥𝐵 𝜒))
 
Theoremsbequbidv 36180* Deduction substituting both sides of a biconditional. (Contributed by GG, 1-Sep-2025.)
(𝜑𝑢 = 𝑣)    &   (𝜑 → (𝜓𝜒))       (𝜑 → ([𝑢 / 𝑥]𝜓 ↔ [𝑣 / 𝑥]𝜒))
 
Theoremdisjeq12dv 36181* Equality theorem for disjoint collection. Deduction version. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → (Disj 𝑥𝐴 𝐶Disj 𝑥𝐵 𝐷))
 
Theoremixpeq12dv 36182* Equality theorem for infinite Cartesian product. Deduction version. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑X𝑥𝐴 𝐶 = X𝑥𝐵 𝐷)
 
Theoremsumeq12sdv 36183* Equality deduction for sum. General version of sumeq2sdv 15751. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → Σ𝑘𝐴 𝐶 = Σ𝑘𝐵 𝐷)
 
Theoremprodeq12sdv 36184* Equality deduction for product. General version of prodeq2sdv 15971. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ∏𝑘𝐴 𝐶 = ∏𝑘𝐵 𝐷)
 
Theoremitgeq12sdv 36185* Equality theorem for an integral. Deduction form. General version of itgeq1d 45878 and itgeq2sdv 36186. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ∫𝐴𝐶 d𝑥 = ∫𝐵𝐷 d𝑥)
 
Theoremitgeq2sdv 36186* Equality theorem for an integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐵 = 𝐶)       (𝜑 → ∫𝐴𝐵 d𝑥 = ∫𝐴𝐶 d𝑥)
 
Theoremditgeq123dv 36187* Equality theorem for the directed integral. Deduction form. General version of ditgeq3sdv 36189. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)    &   (𝜑𝐸 = 𝐹)       (𝜑 → ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]𝐹 d𝑥)
 
Theoremditgeq12d 36188* Equality theorem for the directed integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐴 = 𝐵)    &   (𝜑𝐶 = 𝐷)       (𝜑 → ⨜[𝐴𝐶]𝐸 d𝑥 = ⨜[𝐵𝐷]𝐸 d𝑥)
 
Theoremditgeq3sdv 36189* Equality theorem for the directed integral. Deduction form. (Contributed by GG, 1-Sep-2025.)
(𝜑𝐶 = 𝐷)       (𝜑 → ⨜[𝐴𝐵]𝐶 d𝑥 = ⨜[𝐴𝐵]𝐷 d𝑥)
 
21.12.2  Change bound variables.
 
Theoremin-ax8 36190 A proof of ax-8 2110 that does not rely on ax-8 2110. It employs df-in 3983 to perform alpha-renaming and eliminates disjoint variable conditions using ax-9 2118. Since the nature of this result is unclear, usage of this theorem is discouraged, and this method should not be applied to eliminate axiom dependencies. (Contributed by GG, 1-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
Theoremss-ax8 36191 A proof of ax-8 2110 that does not rely on ax-8 2110. It employs df-ss 3993 to perform alpha-renaming and eliminates disjoint variable conditions using ax-9 2118. Contrary to in-ax8 36190, this proof does not rely on df-cleq 2732, therefore using fewer axioms . This method should not be applied to eliminate axiom dependencies. (Contributed by GG, 30-Aug-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))
 
21.12.2.1  Change bound variables and domains.
 
Theoremcbvralvw2 36192* Change bound variable and domain in the restricted universal quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝐴 𝜑 ↔ ∀𝑦𝐵 𝜓)
 
Theoremcbvrexvw2 36193* Change bound variable and domain in the restricted existential quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝐴 𝜑 ↔ ∃𝑦𝐵 𝜓)
 
Theoremcbvrmovw2 36194* Change bound variable and domain in the restricted at-most-one quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃*𝑥𝐴 𝜑 ↔ ∃*𝑦𝐵 𝜓)
 
Theoremcbvreuvw2 36195* Change bound variable and domain in the restricted existential uniqueness quantifier, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦𝐴 = 𝐵)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∃!𝑥𝐴 𝜑 ↔ ∃!𝑦𝐵 𝜓)
 
Theoremcbvsbcvw2 36196* Change bound variable of a class substitution using implicit substitution. General version of cbvsbcvw 3839. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝑥 = 𝑦 → (𝜑𝜓))       ([𝐴 / 𝑥]𝜑[𝐵 / 𝑦]𝜓)
 
Theoremcbvcsbvw2 36197* Change bound variable of a proper substitution into a class using implicit substitution. General version of cbvcsbv 3933. (Contributed by GG, 1-Sep-2025.)
𝐴 = 𝐵    &   (𝑥 = 𝑦𝐶 = 𝐷)       𝐴 / 𝑥𝐶 = 𝐵 / 𝑦𝐷
 
Theoremcbviunvw2 36198* Change bound variable and domain in indexed unions, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦𝐶 = 𝐷)    &   (𝑥 = 𝑦𝐴 = 𝐵)        𝑥𝐴 𝐶 = 𝑦𝐵 𝐷
 
Theoremcbviinvw2 36199* Change bound variable and domain in an indexed intersection, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦𝐶 = 𝐷)    &   (𝑥 = 𝑦𝐴 = 𝐵)        𝑥𝐴 𝐶 = 𝑦𝐵 𝐷
 
Theoremcbvmptvw2 36200* Change bound variable and domain in a maps-to function, using implicit substitution. (Contributed by GG, 14-Aug-2025.)
(𝑥 = 𝑦𝐶 = 𝐷)    &   (𝑥 = 𝑦𝐴 = 𝐵)       (𝑥𝐴𝐶) = (𝑦𝐵𝐷)
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