P. 360 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 35901-36000   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-rqp 35901*	There is a unique element of (ℤ ↑m (0...(𝑝 − 1))) ~Qp -equivalent to any element of (ℤ ↑m ℤ), if the sequences are zero for sufficiently large negative values; this function selects that element. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ /Qp = (𝑝 ∈ ℙ ↦ (~Qp ∩ ⦋{𝑓 ∈ (ℤ ↑m ℤ) ∣ ∃𝑥 ∈ ran ℤ≥(◡𝑓 “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑦⦌(𝑦 × (𝑦 ∩ (ℤ ↑m (0...(𝑝 − 1)))))))
Definition	df-qp 35902*	Define the 𝑝-adic completion of the rational numbers, as a normed field structure with a total order (that is not compatible with the operations). (Contributed by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 10-Oct-2021.)
⊢ Qp = (𝑝 ∈ ℙ ↦ ⦋{ℎ ∈ (ℤ ↑m (0...(𝑝 − 1))) ∣ ∃𝑥 ∈ ran ℤ≥(◡ℎ “ (ℤ ∖ {0})) ⊆ 𝑥} / 𝑏⦌(({⟨(Base‘ndx), 𝑏⟩, ⟨(+g‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑓 ∘f + 𝑔)))⟩, ⟨(.r‘ndx), (𝑓 ∈ 𝑏, 𝑔 ∈ 𝑏 ↦ ((/Qp‘𝑝)‘(𝑛 ∈ ℤ ↦ Σ𝑘 ∈ ℤ ((𝑓‘𝑘) · (𝑔‘(𝑛 − 𝑘))))))⟩} ∪ {⟨(le‘ndx), {⟨𝑓, 𝑔⟩ ∣ ({𝑓, 𝑔} ⊆ 𝑏 ∧ Σ𝑘 ∈ ℤ ((𝑓‘-𝑘) · ((𝑝 + 1)↑-𝑘)) < Σ𝑘 ∈ ℤ ((𝑔‘-𝑘) · ((𝑝 + 1)↑-𝑘)))}⟩}) toNrmGrp (𝑓 ∈ 𝑏 ↦ if(𝑓 = (ℤ × {0}), 0, (𝑝↑-inf((◡𝑓 “ (ℤ ∖ {0})), ℝ, < ))))))
Definition	df-zp 35903	Define the 𝑝-adic integers, as a subset of the 𝑝-adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ Zp = (ZRing ∘ Qp)
Definition	df-qpa 35904*	Define the completion of the 𝑝-adic rationals. Here we simply define it as the splitting field of a dense sequence of polynomials (using as the 𝑛-th set the collection of polynomials with degree less than 𝑛 and with coefficients < (𝑝↑𝑛)). Krasner's lemma will then show that all monic polynomials have splitting fields isomorphic to a sufficiently close Eisenstein polynomial from the list, and unramified extensions are generated by the polynomial 𝑥↑(𝑝↑𝑛) − 𝑥, which is in the list. Thus, every finite extension of Qp is a subfield of this field extension, so it is algebraically closed. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ _Qp = (𝑝 ∈ ℙ ↦ ⦋(Qp‘𝑝) / 𝑟⦌(𝑟 polySplitLim (𝑛 ∈ ℕ ↦ {𝑓 ∈ (Poly1‘𝑟) ∣ ((𝑟deg1𝑓) ≤ 𝑛 ∧ ∀𝑑 ∈ ran (coe1‘𝑓)(◡𝑑 “ (ℤ ∖ {0})) ⊆ (0...𝑛))})))
Definition	df-cp 35905	Define the metric completion of the algebraic completion of the 𝑝 -adic rationals. (Contributed by Mario Carneiro, 2-Dec-2014.)
⊢ Cp = ( cplMetSp ∘ _Qp)
21.7  Mathbox for Filip Cernatescu I hope someone will enjoy solving (proving) the simple equations, inequalities, and calculations from this mathbox. I have proved these problems (theorems) using the Milpgame proof assistant. (It can be downloaded from https://us.metamath.org/other/milpgame/milpgame.html.)
Theorem	problem1 35906	Practice problem 1. Clues: 5p4e9 12329 3p2e5 12322 eqtri 2764 oveq1i 7369. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ ((3 + 2) + 4) = 9
Theorem	problem2 35907	Practice problem 2. Clues: oveq12i 7371 adddiri 11154 add4i 11367 mulcli 11148 recni 11155 2re 12250 3eqtri 2768 10re 12658 5re 12263 1re 11140 4re 12260 eqcomi 2750 5p4e9 12329 oveq1i 7369 df-3 12240. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Revised by AV, 9-Sep-2021.) (Proof modification is discouraged.)
⊢ (((2 · ;10) + 5) + ((1 · ;10) + 4)) = ((3 · ;10) + 9)
Theorem	problem3 35908	Practice problem 3. Clues: eqcomi 2750 eqtri 2764 subaddrii 11479 recni 11155 4re 12260 3re 12256 1re 11140 df-4 12241 addcomi 11333. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ (𝐴 + 3) = 4    ⇒   ⊢ 𝐴 = 1
Theorem	problem4 35909	Practice problem 4. Clues: pm3.2i 472 eqcomi 2750 eqtri 2764 subaddrii 11479 recni 11155 7re 12269 6re 12266 ax-1cn 11092 df-7 12244 ax-mp 5 oveq1i 7369 3cn 12257 2cn 12251 df-3 12240 mullidi 11146 subdiri 11596 mp3an 1470 mulcli 11148 subadd23 11401 oveq2i 7370 oveq12i 7371 3t2e6 12337 mulcomi 11149 subcli 11466 biimpri 230 subadd2i 11478. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ (𝐴 + 𝐵) = 3    &   ⊢ ((3 · 𝐴) + (2 · 𝐵)) = 7    ⇒   ⊢ (𝐴 = 1 ∧ 𝐵 = 2)
Theorem	problem5 35910	Practice problem 5. Clues: 3brtr3i 5103 mpbi 232 breqtri 5099 ltaddsubi 11707 remulcli 11157 2re 12250 3re 12256 9re 12275 eqcomi 2750 mvlladdi 11408 3cn 6cn 12267 eqtr3i 2766 6p3e9 12331 addcomi 11333 ltdiv1ii 12080 6re 12266 nngt0i 12211 2nn 12249 divcan3i 11896 recni 11155 2cn 12251 2ne0 12280 mpbir 233 eqtri 2764 mulcomi 11149 3t2e6 12337 divmuli 11904. (Contributed by Filip Cernatescu, 16-Mar-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ ℝ    &   ⊢ ((2 · 𝐴) + 3) < 9    ⇒   ⊢ 𝐴 < 3
Theorem	quad3 35911	Variant of quadratic equation with discriminant expanded. (Contributed by Filip Cernatescu, 19-Oct-2019.)
⊢ 𝑋 ∈ ℂ    &   ⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐴 ≠ 0    &   ⊢ 𝐵 ∈ ℂ    &   ⊢ 𝐶 ∈ ℂ    &   ⊢ ((𝐴 · (𝑋↑2)) + ((𝐵 · 𝑋) + 𝐶)) = 0    ⇒   ⊢ (𝑋 = ((-𝐵 + (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)) ∨ 𝑋 = ((-𝐵 − (√‘((𝐵↑2) − (4 · (𝐴 · 𝐶))))) / (2 · 𝐴)))
21.8  Mathbox for Paul Chapman
21.8.1  Real and complex numbers (cont.)
Theorem	climuzcnv 35912*	Utility lemma to convert between 𝑚 ≤ 𝑘 and 𝑘 ∈ (ℤ≥‘𝑚) in limit theorems. (Contributed by Paul Chapman, 10-Nov-2012.)
⊢ (𝑚 ∈ ℕ → ((𝑘 ∈ (ℤ≥‘𝑚) → 𝜑) ↔ (𝑘 ∈ ℕ → (𝑚 ≤ 𝑘 → 𝜑))))
Theorem	sinccvglem 35913*	((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Revised by Mario Carneiro, 21-May-2014.)
⊢ (𝜑 → 𝐹:ℕ⟶(ℝ ∖ {0}))    &   ⊢ (𝜑 → 𝐹 ⇝ 0)    &   ⊢ 𝐺 = (𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥))    &   ⊢ 𝐻 = (𝑥 ∈ ℂ ↦ (1 − ((𝑥↑2) / 3)))    &   ⊢ (𝜑 → 𝑀 ∈ ℕ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ≥‘𝑀)) → (abs‘(𝐹‘𝑘)) < 1)    ⇒   ⊢ (𝜑 → (𝐺 ∘ 𝐹) ⇝ 1)
Theorem	sinccvg 35914*	((sin‘𝑥) / 𝑥) ⇝ 1 as (real) 𝑥 ⇝ 0. (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
⊢ ((𝐹:ℕ⟶(ℝ ∖ {0}) ∧ 𝐹 ⇝ 0) → ((𝑥 ∈ (ℝ ∖ {0}) ↦ ((sin‘𝑥) / 𝑥)) ∘ 𝐹) ⇝ 1)
Theorem	circum 35915*	The circumference of a circle of radius 𝑅, defined as the limit as 𝑛 ⇝ +∞ of the perimeter of an inscribed n-sided isogons, is ((2 · π) · 𝑅). (Contributed by Paul Chapman, 10-Nov-2012.) (Proof shortened by Mario Carneiro, 21-May-2014.)
⊢ 𝐴 = ((2 · π) / 𝑛)    &   ⊢ 𝑃 = (𝑛 ∈ ℕ ↦ ((2 · 𝑛) · (𝑅 · (sin‘(𝐴 / 2)))))    &   ⊢ 𝑅 ∈ ℝ    ⇒   ⊢ 𝑃 ⇝ ((2 · π) · 𝑅)
21.8.2  Miscellaneous theorems
Theorem	elfzm12 35916	Membership in a curtailed finite sequence of integers. (Contributed by Paul Chapman, 17-Nov-2012.)
⊢ (𝑁 ∈ ℕ → (𝑀 ∈ (1...(𝑁 − 1)) → 𝑀 ∈ (1...𝑁)))
Theorem	nn0seqcvg 35917*	A strictly-decreasing nonnegative integer sequence with initial term 𝑁 reaches zero by the 𝑁 th term. Inference version. (Contributed by Paul Chapman, 31-Mar-2011.)
⊢ 𝐹:ℕ0⟶ℕ0    &   ⊢ 𝑁 = (𝐹‘0)    &   ⊢ (𝑘 ∈ ℕ0 → ((𝐹‘(𝑘 + 1)) ≠ 0 → (𝐹‘(𝑘 + 1)) < (𝐹‘𝑘)))    ⇒   ⊢ (𝐹‘𝑁) = 0
Theorem	lediv2aALT 35918	Division of both sides of 'less than or equal to' by a nonnegative number. (Contributed by Paul Chapman, 7-Sep-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (((𝐴 ∈ ℝ ∧ 0 < 𝐴) ∧ (𝐵 ∈ ℝ ∧ 0 < 𝐵) ∧ (𝐶 ∈ ℝ ∧ 0 ≤ 𝐶)) → (𝐴 ≤ 𝐵 → (𝐶 / 𝐵) ≤ (𝐶 / 𝐴)))
Theorem	abs2sqlei 35919	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2))
Theorem	abs2sqlti 35920	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2))
Theorem	abs2sqle 35921	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) ≤ (abs‘𝐵) ↔ ((abs‘𝐴)↑2) ≤ ((abs‘𝐵)↑2)))
Theorem	abs2sqlt 35922	The absolute values of two numbers compare as their squares. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((abs‘𝐴) < (abs‘𝐵) ↔ ((abs‘𝐴)↑2) < ((abs‘𝐵)↑2)))
Theorem	abs2difi 35923	Difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ ((abs‘𝐴) − (abs‘𝐵)) ≤ (abs‘(𝐴 − 𝐵))
Theorem	abs2difabsi 35924	Absolute value of difference of absolute values. (Contributed by Paul Chapman, 7-Sep-2007.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (abs‘((abs‘𝐴) − (abs‘𝐵))) ≤ (abs‘(𝐴 − 𝐵))
21.9  Mathbox for Hongxiu Chen
Theorem	2thALT 35925	Alternate proof of 2th 266. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	orbi2iALT 35926	Alternate proof of orbi2i 919. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ ((𝜒 ∨ 𝜑) ↔ (𝜒 ∨ 𝜓))
Theorem	pm3.48ALT 35927	Alternate proof of pm3.48 972. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (((𝜑 → 𝜓) ∧ (𝜒 → 𝜃)) → ((𝜑 ∨ 𝜒) → (𝜓 ∨ 𝜃)))
Theorem	3jcadALT 35928	Alternate proof of 3jcad 1136. (Contributed by Hongxiu Chen, 29-Jun-2025.) (Proof modification is discouraged.) Use 3jcad instead. (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 → 𝜃))    &   ⊢ (𝜑 → (𝜓 → 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ∧ 𝜃 ∧ 𝜏)))
21.10  Mathbox for Adrian Ducourtial
21.10.1  Propositional calculus
Theorem	currybi 35929	Biconditional version of Curry's paradox. If some proposition 𝜑 amounts to the self-referential statement "This very statement is equivalent to 𝜓", then 𝜓 is true. See bj-currypara 36883 in BJ's mathbox for the classical version. (Contributed by Adrian Ducourtial, 18-Mar-2025.)
⊢ ((𝜑 ↔ (𝜑 ↔ 𝜓)) → 𝜓)
Theorem	antnest 35930	Suppose 𝜑, 𝜓 are distinct atomic propositional formulas, and let Γ be the smallest class of formulas for which ⊤ ∈ Γ and (𝜒 → 𝜑), (𝜒 → 𝜓) ∈ Γ for 𝜒 ∈ Γ. The present theorem is then an element of Γ, and the implications occurring in the theorem are in one-to-one correspondence with the formulas in Γ up to logical equivalence. In particular, the theorem itself is equivalent to ⊤ ∈ Γ. (Contributed by Adrian Ducourtial, 2-Oct-2025.)
⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
Theorem	antnestlaw3lem 35931	Lemma for antnestlaw3 35934. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ (¬ (((𝜑 → 𝜓) → 𝜒) → 𝜒) → ¬ (((𝜑 → 𝜒) → 𝜓) → 𝜓))
Theorem	antnestlaw1 35932	A law of nested antecedents. The converse direction is a subschema of pm2.27 42. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜓) ↔ (𝜑 → 𝜓))
Theorem	antnestlaw2 35933	A law of nested antecedents. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ ((((𝜑 → 𝜓) → 𝜓) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜒))
Theorem	antnestlaw3 35934	A law of nested antecedents. Compare with looinv 205. (Contributed by Adrian Ducourtial, 5-Dec-2025.)
⊢ ((((𝜑 → 𝜓) → 𝜒) → 𝜒) ↔ (((𝜑 → 𝜒) → 𝜓) → 𝜓))
Theorem	antnestALT 35935	Alternative proof of antnest 35930 from the valid schema ((((⊤ → 𝜑) → 𝜑) → 𝜓) → 𝜓) using laws of nested antecedents. Our proof uses only the laws antnestlaw1 35932 and antnestlaw3 35934. (Contributed by Adrian Ducourtial, 5-Dec-2025.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((((((⊤ → 𝜑) → 𝜓) → 𝜓) → 𝜑) → 𝜓) → 𝜓)
21.10.2  Clone theory
Syntax	ccloneop 35936	Syntax for the function of the class of operations on a set.
class CloneOp
Definition	df-cloneop 35937*	Define the function that sends a set to the class of clone-theoretic operations on the set. For convenience, we take an operation on 𝑎 to be a function on finite sequences of elements of 𝑎 (rather than tuples) with values in 𝑎. Following line 6 of [Szendrei] p. 11, the arity 𝑛 of an operation (here, the length of the sequences at which the operation is defined) is always finite and nonzero, whence 𝑛 is taken to be a nonzero finite ordinal. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ CloneOp = (𝑎 ∈ V ↦ {𝑥 ∣ ∃𝑛 ∈ (ω ∖ 1o)𝑥 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛))})
Syntax	cprj 35938	Syntax for the function of projections on sets.
class prj
Definition	df-prj 35939*	Define the function that, for a set 𝑎, arity 𝑛, and index 𝑖, returns the 𝑖-th 𝑛-ary projection on 𝑎. This is the 𝑛-ary operation on 𝑎 that, for any sequence of 𝑛 elements of 𝑎, returns the element having index 𝑖. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ prj = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑖 ∈ 𝑛 ↦ (𝑥 ∈ (𝑎 ↑m 𝑛) ↦ (𝑥‘𝑖))))
Syntax	csuppos 35940	Syntax for the function of superpositions.
class suppos
Definition	df-suppos 35941*	Define the function that, when given an 𝑛-ary operation 𝑓 and 𝑛 many 𝑚-ary operations (𝑔‘∅), ..., (𝑔‘∪ 𝑛), returns the superposition of 𝑓 with the (𝑔‘𝑖), itself another 𝑚-ary operation on 𝑎. Given 𝑥 (a sequence of 𝑚 arguments in 𝑎), the superposition effectively applies each of the (𝑔‘𝑖) to 𝑥, then applies 𝑓 to the resulting sequence of 𝑛 function values. This can be seen as a generalized version of function composition; see paragraph 3 of [Szendrei] p. 11. (Contributed by Adrian Ducourtial, 3-Apr-2025.)
⊢ suppos = (𝑎 ∈ V ↦ (𝑛 ∈ (ω ∖ 1o), 𝑚 ∈ (ω ∖ 1o) ↦ (𝑓 ∈ (𝑎 ↑m (𝑎 ↑m 𝑛)), 𝑔 ∈ ((𝑎 ↑m (𝑎 ↑m 𝑚)) ↑m 𝑛) ↦ (𝑥 ∈ (𝑎 ↑m 𝑚) ↦ (𝑓‘(𝑖 ∈ 𝑛 ↦ ((𝑔‘𝑖)‘𝑥)))))))
21.11  Mathbox for Scott Fenton
21.11.1  ZFC Axioms in primitive form
Theorem	axextprim 35942	ax-ext 2713 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ((𝑥 ∈ 𝑦 → 𝑥 ∈ 𝑧) → ((𝑥 ∈ 𝑧 → 𝑥 ∈ 𝑦) → 𝑦 = 𝑧))
Theorem	axrepprim 35943	ax-rep 5201 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ (¬ ∀𝑦 ¬ ∀𝑧(𝜑 → 𝑧 = 𝑦) → ∀𝑧 ¬ ((∀𝑦 𝑧 ∈ 𝑥 → ¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑)) → ¬ (¬ ∀𝑥(∀𝑧 𝑥 ∈ 𝑦 → ¬ ∀𝑦𝜑) → ∀𝑦 𝑧 ∈ 𝑥)))
Theorem	axunprim 35944	ax-un 7681 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ∀𝑦(¬ ∀𝑥(𝑦 ∈ 𝑥 → ¬ 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥)
Theorem	axpowprim 35945	ax-pow 5296 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ (∀𝑥 ¬ ∀𝑦(∀𝑥(¬ ∀𝑧 ¬ 𝑥 ∈ 𝑦 → ∀𝑦 𝑥 ∈ 𝑧) → 𝑦 ∈ 𝑥) → 𝑥 = 𝑦)
Theorem	axregprim 35946	ax-reg 9501 without distinct variable conditions or defined symbols. (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ (𝑥 ∈ 𝑦 → ¬ ∀𝑥(𝑥 ∈ 𝑦 → ¬ ∀𝑧(𝑧 ∈ 𝑥 → ¬ 𝑧 ∈ 𝑦)))
Theorem	axinfprim 35947	ax-inf 9554 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 13-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ (𝑦 ∈ 𝑥 → ¬ ∀𝑦(𝑦 ∈ 𝑥 → ¬ ∀𝑧(𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑥))))
Theorem	axacprim 35948	ax-ac 10377 without distinct variable conditions or defined symbols. (New usage is discouraged.) (Contributed by Scott Fenton, 26-Oct-2010.)
⊢ ¬ ∀𝑥 ¬ ∀𝑦∀𝑧(∀𝑥 ¬ (𝑦 ∈ 𝑧 → ¬ 𝑧 ∈ 𝑤) → ¬ ∀𝑤 ¬ ∀𝑦 ¬ ((¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))) → 𝑦 = 𝑤) → ¬ (𝑦 = 𝑤 → ¬ ∀𝑤(𝑦 ∈ 𝑧 → (𝑧 ∈ 𝑤 → (𝑦 ∈ 𝑤 → ¬ 𝑤 ∈ 𝑥))))))
21.11.2  Untangled classes
Theorem	untelirr 35949*	We call a class "untanged" if all its members are not members of themselves. The term originates from Isbell (see citation in dfon2 36031). Using this concept, we can avoid a lot of the uses of the Axiom of Regularity. Here, we prove a series of properties of untanged classes. First, we prove that an untangled class is not a member of itself. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ¬ 𝐴 ∈ 𝐴)
Theorem	untuni 35950*	The union of a class is untangled iff all its members are untangled. (Contributed by Scott Fenton, 28-Feb-2011.)
⊢ (∀𝑥 ∈ ∪ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑦 ∈ 𝐴 ∀𝑥 ∈ 𝑦 ¬ 𝑥 ∈ 𝑥)
Theorem	untsucf 35951*	If a class is untangled, then so is its successor. (Contributed by Scott Fenton, 28-Feb-2011.) (Revised by Mario Carneiro, 11-Dec-2016.)
⊢ Ⅎ𝑦𝐴    ⇒   ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 → ∀𝑦 ∈ suc 𝐴 ¬ 𝑦 ∈ 𝑦)
Theorem	unt0 35952	The null set is untangled. (Contributed by Scott Fenton, 10-Mar-2011.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
⊢ ∀𝑥 ∈ ∅ ¬ 𝑥 ∈ 𝑥
Theorem	untint 35953*	If there is an untangled element of a class, then the intersection of the class is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦 → ∀𝑦 ∈ ∩ 𝐴 ¬ 𝑦 ∈ 𝑦)
Theorem	efrunt 35954*	If 𝐴 is well-founded by E, then it is untangled. (Contributed by Scott Fenton, 1-Mar-2011.)
⊢ ( E Fr 𝐴 → ∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥)
Theorem	untangtr 35955*	A transitive class is untangled iff its elements are. (Contributed by Scott Fenton, 7-Mar-2011.)
⊢ (Tr 𝐴 → (∀𝑥 ∈ 𝐴 ¬ 𝑥 ∈ 𝑥 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝑥 ¬ 𝑦 ∈ 𝑦))
21.11.3  Extra propositional calculus theorems
Theorem	3jaodd 35956	Double deduction form of 3jaoi 1437. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜂)))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜂)))    &   ⊢ (𝜑 → (𝜓 → (𝜏 → 𝜂)))    ⇒   ⊢ (𝜑 → (𝜓 → ((𝜒 ∨ 𝜃 ∨ 𝜏) → 𝜂)))
Theorem	3orit 35957	Closed form of 3ori 1433. (Contributed by Scott Fenton, 20-Apr-2011.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ ((¬ 𝜑 ∧ ¬ 𝜓) → 𝜒))
Theorem	biimpexp 35958	A biconditional in the antecedent is the same as two implications. (Contributed by Scott Fenton, 12-Dec-2010.)
⊢ (((𝜑 ↔ 𝜓) → 𝜒) ↔ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜒)))
21.11.4  Misc. Useful Theorems
Theorem	nepss 35959	Two classes are unequal iff their intersection is a proper subset of one of them. (Contributed by Scott Fenton, 23-Feb-2011.)
⊢ (𝐴 ≠ 𝐵 ↔ ((𝐴 ∩ 𝐵) ⊊ 𝐴 ∨ (𝐴 ∩ 𝐵) ⊊ 𝐵))
Theorem	3ccased 35960	Triple disjunction form of ccased 1045. (Contributed by Scott Fenton, 27-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ (𝜑 → ((𝜒 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜒 ∧ 𝜎) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜃 ∧ 𝜎) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜂) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜁) → 𝜓))    &   ⊢ (𝜑 → ((𝜏 ∧ 𝜎) → 𝜓))    ⇒   ⊢ (𝜑 → (((𝜒 ∨ 𝜃 ∨ 𝜏) ∧ (𝜂 ∨ 𝜁 ∨ 𝜎)) → 𝜓))
Theorem	dfso3 35961*	Expansion of the definition of a strict order. (Contributed by Scott Fenton, 6-Jun-2016.)
⊢ (𝑅 Or 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ∧ (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
Theorem	brtpid1 35962	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{⟨𝐴, 𝐵⟩, 𝐶, 𝐷}𝐵
Theorem	brtpid2 35963	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{𝐶, ⟨𝐴, 𝐵⟩, 𝐷}𝐵
Theorem	brtpid3 35964	A binary relation involving unordered triples. (Contributed by Scott Fenton, 7-Jun-2016.)
⊢ 𝐴{𝐶, 𝐷, ⟨𝐴, 𝐵⟩}𝐵
Theorem	iota5f 35965*	A method for computing iota. (Contributed by Scott Fenton, 13-Dec-2017.)
⊢ Ⅎ𝑥𝜑    &   ⊢ Ⅎ𝑥𝐴    &   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (𝜓 ↔ 𝑥 = 𝐴))    ⇒   ⊢ ((𝜑 ∧ 𝐴 ∈ 𝑉) → (℩𝑥𝜓) = 𝐴)
Theorem	jath 35966	Closed form of ja 187. Proved using the completeness script. (Proof modification is discouraged.) (Contributed by Scott Fenton, 13-Dec-2021.)
⊢ ((¬ 𝜑 → 𝜒) → ((𝜓 → 𝜒) → ((𝜑 → 𝜓) → 𝜒)))
Theorem	xpab 35967*	Cartesian product of two class abstractions. (Contributed by Scott Fenton, 19-Aug-2024.)
⊢ ({𝑥 ∣ 𝜑} × {𝑦 ∣ 𝜓}) = {⟨𝑥, 𝑦⟩ ∣ (𝜑 ∧ 𝜓)}
Theorem	nnuni 35968	The union of a finite ordinal is a finite ordinal. (Contributed by Scott Fenton, 17-Oct-2024.)
⊢ (𝐴 ∈ ω → ∪ 𝐴 ∈ ω)
21.11.5  Properties of real and complex numbers
Theorem	sqdivzi 35969	Distribution of square over division. (Contributed by Scott Fenton, 7-Jun-2013.)
⊢ 𝐴 ∈ ℂ    &   ⊢ 𝐵 ∈ ℂ    ⇒   ⊢ (𝐵 ≠ 0 → ((𝐴 / 𝐵)↑2) = ((𝐴↑2) / (𝐵↑2)))
Theorem	supfz 35970	The supremum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → sup((𝑀...𝑁), ℤ, < ) = 𝑁)
Theorem	inffz 35971	The infimum of a finite sequence of integers. (Contributed by Scott Fenton, 8-Aug-2013.) (Revised by AV, 10-Oct-2021.)
⊢ (𝑁 ∈ (ℤ≥‘𝑀) → inf((𝑀...𝑁), ℤ, < ) = 𝑀)
Theorem	fz0n 35972	The sequence (0...(𝑁 − 1)) is empty iff 𝑁 is zero. (Contributed by Scott Fenton, 16-May-2014.)
⊢ (𝑁 ∈ ℕ0 → ((0...(𝑁 − 1)) = ∅ ↔ 𝑁 = 0))
Theorem	shftvalg 35973	Value of a sequence shifted by 𝐴. (Contributed by Scott Fenton, 16-Dec-2017.)
⊢ ((𝐹 ∈ 𝑉 ∧ 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → ((𝐹 shift 𝐴)‘𝐵) = (𝐹‘(𝐵 − 𝐴)))
Theorem	divcnvlin 35974*	Limit of the ratio of two linear functions. (Contributed by Scott Fenton, 17-Dec-2017.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐴 ∈ ℂ)    &   ⊢ (𝜑 → 𝐵 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹 ∈ 𝑉)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵)))    ⇒   ⊢ (𝜑 → 𝐹 ⇝ 1)
Theorem	climlec3 35975*	Comparison of a constant to the limit of a sequence. (Contributed by Scott Fenton, 5-Jan-2018.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐵 ∈ ℝ)    &   ⊢ (𝜑 → 𝐹 ⇝ 𝐴)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ≤ 𝐵)    ⇒   ⊢ (𝜑 → 𝐴 ≤ 𝐵)
Theorem	iexpire 35976	i raised to itself is real. (Contributed by Scott Fenton, 13-Apr-2020.)
⊢ (i↑𝑐i) ∈ ℝ
Theorem	bcneg1 35977	The binomial coefficient over negative one is zero. (Contributed by Scott Fenton, 29-May-2020.)
⊢ (𝑁 ∈ ℕ0 → (𝑁C-1) = 0)
Theorem	bcm1nt 35978	The proportion of one binomial coefficient to another with 𝑁 decreased by 1. (Contributed by Scott Fenton, 23-Jun-2020.)
⊢ ((𝑁 ∈ ℕ ∧ 𝐾 ∈ (0...(𝑁 − 1))) → (𝑁C𝐾) = (((𝑁 − 1)C𝐾) · (𝑁 / (𝑁 − 𝐾))))
Theorem	bcprod 35979*	A product identity for binomial coefficients. (Contributed by Scott Fenton, 23-Jun-2020.)
⊢ (𝑁 ∈ ℕ → ∏𝑘 ∈ (1...(𝑁 − 1))((𝑁 − 1)C𝑘) = ∏𝑘 ∈ (1...(𝑁 − 1))(𝑘↑((2 · 𝑘) − 𝑁)))
Theorem	bccolsum 35980*	A column-sum rule for binomial coefficients. (Contributed by Scott Fenton, 24-Jun-2020.)
⊢ ((𝑁 ∈ ℕ0 ∧ 𝐶 ∈ ℕ0) → Σ𝑘 ∈ (0...𝑁)(𝑘C𝐶) = ((𝑁 + 1)C(𝐶 + 1)))
21.11.6  Infinite products
Theorem	iprodefisumlem 35981	Lemma for iprodefisum 35982. (Contributed by Scott Fenton, 11-Feb-2018.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ (𝜑 → 𝐹:𝑍⟶ℂ)    ⇒   ⊢ (𝜑 → seq𝑀( · , (exp ∘ 𝐹)) = (exp ∘ seq𝑀( + , 𝐹)))
Theorem	iprodefisum 35982*	Applying the exponential function to an infinite sum yields an infinite product. (Contributed by Scott Fenton, 11-Feb-2018.)
⊢ 𝑍 = (ℤ≥‘𝑀)    &   ⊢ (𝜑 → 𝑀 ∈ ℤ)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐵)    &   ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ)    &   ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ )    ⇒   ⊢ (𝜑 → ∏𝑘 ∈ 𝑍 (exp‘𝐵) = (exp‘Σ𝑘 ∈ 𝑍 𝐵))
Theorem	iprodgam 35983*	An infinite product version of Euler's gamma function. (Contributed by Scott Fenton, 12-Feb-2018.)
⊢ (𝜑 → 𝐴 ∈ (ℂ ∖ (ℤ ∖ ℕ)))    ⇒   ⊢ (𝜑 → (Γ‘𝐴) = (∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝑐𝐴) / (1 + (𝐴 / 𝑘))) / 𝐴))
21.11.7  Factorial limits
Theorem	faclimlem1 35984*	Lemma for faclim 35987. Closed form for a particular sequence. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) = (𝑥 ∈ ℕ ↦ ((𝑀 + 1) · ((𝑥 + 1) / (𝑥 + (𝑀 + 1))))))
Theorem	faclimlem2 35985*	Lemma for faclim 35987. Show a limit for the inductive step. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ (𝑀 ∈ ℕ0 → seq1( · , (𝑛 ∈ ℕ ↦ (((1 + (𝑀 / 𝑛)) · (1 + (1 / 𝑛))) / (1 + ((𝑀 + 1) / 𝑛))))) ⇝ (𝑀 + 1))
Theorem	faclimlem3 35986	Lemma for faclim 35987. Algebraic manipulation for the final induction. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ ((𝑀 ∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (((1 + (1 / 𝐵))↑(𝑀 + 1)) / (1 + ((𝑀 + 1) / 𝐵))) = ((((1 + (1 / 𝐵))↑𝑀) / (1 + (𝑀 / 𝐵))) · (((1 + (𝑀 / 𝐵)) · (1 + (1 / 𝐵))) / (1 + ((𝑀 + 1) / 𝐵)))))
Theorem	faclim 35987*	An infinite product expression relating to factorials. Originally due to Euler. (Contributed by Scott Fenton, 22-Nov-2017.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((1 + (1 / 𝑛))↑𝐴) / (1 + (𝐴 / 𝑛))))    ⇒   ⊢ (𝐴 ∈ ℕ0 → seq1( · , 𝐹) ⇝ (!‘𝐴))
Theorem	iprodfac 35988*	An infinite product expression for factorial. (Contributed by Scott Fenton, 15-Dec-2017.)
⊢ (𝐴 ∈ ℕ0 → (!‘𝐴) = ∏𝑘 ∈ ℕ (((1 + (1 / 𝑘))↑𝐴) / (1 + (𝐴 / 𝑘))))
Theorem	faclim2 35989*	Another factorial limit due to Euler. (Contributed by Scott Fenton, 17-Dec-2017.)
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (((!‘𝑛) · ((𝑛 + 1)↑𝑀)) / (!‘(𝑛 + 𝑀))))    ⇒   ⊢ (𝑀 ∈ ℕ0 → 𝐹 ⇝ 1)
21.11.8  Greatest common divisor and divisibility
Theorem	gcd32 35990	Swap the second and third arguments of a gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐶 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐶) = ((𝐴 gcd 𝐶) gcd 𝐵))
Theorem	gcdabsorb 35991	Absorption law for gcd. (Contributed by Scott Fenton, 8-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.)
⊢ ((𝐴 ∈ ℤ ∧ 𝐵 ∈ ℤ) → ((𝐴 gcd 𝐵) gcd 𝐵) = (𝐴 gcd 𝐵))
21.11.9  Properties of relationships
Theorem	dftr6 35992	A potential definition of transitivity for sets. (Contributed by Scott Fenton, 18-Mar-2012.)
⊢ 𝐴 ∈ V    ⇒   ⊢ (Tr 𝐴 ↔ 𝐴 ∈ (V ∖ ran (( E ∘ E ) ∖ E )))
Theorem	coep 35993*	Composition with the membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴( E ∘ 𝑅)𝐵 ↔ ∃𝑥 ∈ 𝐵 𝐴𝑅𝑥)
Theorem	coepr 35994*	Composition with the converse membership relation. (Contributed by Scott Fenton, 18-Feb-2013.)
⊢ 𝐴 ∈ V    &   ⊢ 𝐵 ∈ V    ⇒   ⊢ (𝐴(𝑅 ∘ ◡ E )𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑥𝑅𝐵)
Theorem	dffr5 35995	A quantifier-free definition of a well-founded relationship. (Contributed by Scott Fenton, 11-Apr-2011.)
⊢ (𝑅 Fr 𝐴 ↔ (𝒫 𝐴 ∖ {∅}) ⊆ ran ( E ∖ ( E ∘ ◡𝑅)))
Theorem	dfso2 35996	Quantifier-free definition of a strict order. (Contributed by Scott Fenton, 22-Feb-2013.)
⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ (𝐴 × 𝐴) ⊆ (𝑅 ∪ ( I ∪ ◡𝑅))))
Theorem	br8 35997*	Substitution for an eight-place predicate. (Contributed by Scott Fenton, 26-Sep-2013.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑔 = 𝐺 → (𝜁 ↔ 𝜎))    &   ⊢ (ℎ = 𝐻 → (𝜎 ↔ 𝜌))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 ∃𝑔 ∈ 𝑃 ∃ℎ ∈ 𝑃 (𝑝 = ⟨⟨𝑎, 𝑏⟩, ⟨𝑐, 𝑑⟩⟩ ∧ 𝑞 = ⟨⟨𝑒, 𝑓⟩, ⟨𝑔, ℎ⟩⟩ ∧ 𝜑)}    ⇒   ⊢ (((𝑋 ∈ 𝑆 ∧ 𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄) ∧ (𝐹 ∈ 𝑄 ∧ 𝐺 ∈ 𝑄 ∧ 𝐻 ∈ 𝑄)) → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩𝑅⟨⟨𝐸, 𝐹⟩, ⟨𝐺, 𝐻⟩⟩ ↔ 𝜌))
Theorem	br6 35998*	Substitution for a six-place predicate. (Contributed by Scott Fenton, 4-Oct-2013.) (Revised by Mario Carneiro, 3-May-2015.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑒 = 𝐸 → (𝜏 ↔ 𝜂))    &   ⊢ (𝑓 = 𝐹 → (𝜂 ↔ 𝜁))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 ∃𝑒 ∈ 𝑃 ∃𝑓 ∈ 𝑃 (𝑝 = ⟨𝑎, ⟨𝑏, 𝑐⟩⟩ ∧ 𝑞 = ⟨𝑑, ⟨𝑒, 𝑓⟩⟩ ∧ 𝜑)}    ⇒   ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄 ∧ 𝐶 ∈ 𝑄) ∧ (𝐷 ∈ 𝑄 ∧ 𝐸 ∈ 𝑄 ∧ 𝐹 ∈ 𝑄)) → (⟨𝐴, ⟨𝐵, 𝐶⟩⟩𝑅⟨𝐷, ⟨𝐸, 𝐹⟩⟩ ↔ 𝜁))
Theorem	br4 35999*	Substitution for a four-place predicate. (Contributed by Scott Fenton, 9-Oct-2013.) (Revised by Mario Carneiro, 14-Oct-2013.)
⊢ (𝑎 = 𝐴 → (𝜑 ↔ 𝜓))    &   ⊢ (𝑏 = 𝐵 → (𝜓 ↔ 𝜒))    &   ⊢ (𝑐 = 𝐶 → (𝜒 ↔ 𝜃))    &   ⊢ (𝑑 = 𝐷 → (𝜃 ↔ 𝜏))    &   ⊢ (𝑥 = 𝑋 → 𝑃 = 𝑄)    &   ⊢ 𝑅 = {⟨𝑝, 𝑞⟩ ∣ ∃𝑥 ∈ 𝑆 ∃𝑎 ∈ 𝑃 ∃𝑏 ∈ 𝑃 ∃𝑐 ∈ 𝑃 ∃𝑑 ∈ 𝑃 (𝑝 = ⟨𝑎, 𝑏⟩ ∧ 𝑞 = ⟨𝑐, 𝑑⟩ ∧ 𝜑)}    ⇒   ⊢ ((𝑋 ∈ 𝑆 ∧ (𝐴 ∈ 𝑄 ∧ 𝐵 ∈ 𝑄) ∧ (𝐶 ∈ 𝑄 ∧ 𝐷 ∈ 𝑄)) → (⟨𝐴, 𝐵⟩𝑅⟨𝐶, 𝐷⟩ ↔ 𝜏))
Theorem	cnvco1 36000	Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
⊢ ◡(◡𝐴 ∘ 𝐵) = (◡𝐵 ∘ 𝐴)