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Theorem List for Metamath Proof Explorer - 37201-37300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoreminidl 37201 The intersection of two ideals is an ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…) ∧ 𝐽 ∈ (Idlβ€˜π‘…)) β†’ (𝐼 ∩ 𝐽) ∈ (Idlβ€˜π‘…))
 
Theoremunichnidl 37202* The union of a nonempty chain of ideals is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
((𝑅 ∈ RingOps ∧ (𝐢 β‰  βˆ… ∧ 𝐢 βŠ† (Idlβ€˜π‘…) ∧ βˆ€π‘– ∈ 𝐢 βˆ€π‘— ∈ 𝐢 (𝑖 βŠ† 𝑗 ∨ 𝑗 βŠ† 𝑖))) β†’ βˆͺ 𝐢 ∈ (Idlβ€˜π‘…))
 
Theoremkeridl 37203 The kernel of a ring homomorphism is an ideal. (Contributed by Jeff Madsen, 3-Jan-2011.)
𝐺 = (1st β€˜π‘†)    &   π‘ = (GIdβ€˜πΊ)    β‡’   ((𝑅 ∈ RingOps ∧ 𝑆 ∈ RingOps ∧ 𝐹 ∈ (𝑅 RingOpsHom 𝑆)) β†’ (◑𝐹 β€œ {𝑍}) ∈ (Idlβ€˜π‘…))
 
Theorempridlval 37204* The class of prime ideals of a ring 𝑅. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ RingOps β†’ (PrIdlβ€˜π‘…) = {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑖 β†’ (π‘Ž βŠ† 𝑖 ∨ 𝑏 βŠ† 𝑖)))})
 
Theoremispridl 37205* The predicate "is a prime ideal". (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ RingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ (Idlβ€˜π‘…)βˆ€π‘ ∈ (Idlβ€˜π‘…)(βˆ€π‘₯ ∈ π‘Ž βˆ€π‘¦ ∈ 𝑏 (π‘₯𝐻𝑦) ∈ 𝑃 β†’ (π‘Ž βŠ† 𝑃 ∨ 𝑏 βŠ† 𝑃)))))
 
Theorempridlidl 37206 A prime ideal is an ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ 𝑃 ∈ (Idlβ€˜π‘…))
 
Theorempridlnr 37207 A prime ideal is a proper ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) β†’ 𝑃 β‰  𝑋)
 
Theorempridl 37208* The main property of a prime ideal. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐻 = (2nd β€˜π‘…)    β‡’   (((𝑅 ∈ RingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (Idlβ€˜π‘…) ∧ 𝐡 ∈ (Idlβ€˜π‘…) ∧ βˆ€π‘₯ ∈ 𝐴 βˆ€π‘¦ ∈ 𝐡 (π‘₯𝐻𝑦) ∈ 𝑃)) β†’ (𝐴 βŠ† 𝑃 ∨ 𝐡 βŠ† 𝑃))
 
Theoremispridl2 37209* A condition that shows an ideal is prime. For commutative rings, this is often taken to be the definition. See ispridlc 37241 for the equivalence in the commutative case. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))) β†’ 𝑃 ∈ (PrIdlβ€˜π‘…))
 
Theoremmaxidlval 37210* The set of maximal ideals of a ring. (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ RingOps β†’ (MaxIdlβ€˜π‘…) = {𝑖 ∈ (Idlβ€˜π‘…) ∣ (𝑖 β‰  𝑋 ∧ βˆ€π‘— ∈ (Idlβ€˜π‘…)(𝑖 βŠ† 𝑗 β†’ (𝑗 = 𝑖 ∨ 𝑗 = 𝑋)))})
 
Theoremismaxidl 37211* The predicate "is a maximal ideal". (Contributed by Jeff Madsen, 5-Jan-2011.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ RingOps β†’ (𝑀 ∈ (MaxIdlβ€˜π‘…) ↔ (𝑀 ∈ (Idlβ€˜π‘…) ∧ 𝑀 β‰  𝑋 ∧ βˆ€π‘— ∈ (Idlβ€˜π‘…)(𝑀 βŠ† 𝑗 β†’ (𝑗 = 𝑀 ∨ 𝑗 = 𝑋)))))
 
Theoremmaxidlidl 37212 A maximal ideal is an ideal. (Contributed by Jeff Madsen, 5-Jan-2011.)
((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdlβ€˜π‘…)) β†’ 𝑀 ∈ (Idlβ€˜π‘…))
 
Theoremmaxidlnr 37213 A maximal ideal is proper. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdlβ€˜π‘…)) β†’ 𝑀 β‰  𝑋)
 
Theoremmaxidlmax 37214 A maximal ideal is a maximal proper ideal. (Contributed by Jeff Madsen, 16-Jun-2011.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdlβ€˜π‘…)) ∧ (𝐼 ∈ (Idlβ€˜π‘…) ∧ 𝑀 βŠ† 𝐼)) β†’ (𝐼 = 𝑀 ∨ 𝐼 = 𝑋))
 
Theoremmaxidln1 37215 One is not contained in any maximal ideal. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐻 = (2nd β€˜π‘…)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdlβ€˜π‘…)) β†’ Β¬ π‘ˆ ∈ 𝑀)
 
Theoremmaxidln0 37216 A ring with a maximal ideal is not the zero ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘ = (GIdβ€˜πΊ)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝑅 ∈ RingOps ∧ 𝑀 ∈ (MaxIdlβ€˜π‘…)) β†’ π‘ˆ β‰  𝑍)
 
21.22.21  Prime rings and integral domains
 
Syntaxcprrng 37217 Extend class notation with the class of prime rings.
class PrRing
 
Syntaxcdmn 37218 Extend class notation with the class of domains.
class Dmn
 
Definitiondf-prrngo 37219 Define the class of prime rings. A ring is prime if the zero ideal is a prime ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
PrRing = {π‘Ÿ ∈ RingOps ∣ {(GIdβ€˜(1st β€˜π‘Ÿ))} ∈ (PrIdlβ€˜π‘Ÿ)}
 
Definitiondf-dmn 37220 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
Dmn = (PrRing ∩ Com2)
 
Theoremisprrngo 37221 The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π‘ = (GIdβ€˜πΊ)    β‡’   (𝑅 ∈ PrRing ↔ (𝑅 ∈ RingOps ∧ {𝑍} ∈ (PrIdlβ€˜π‘…)))
 
Theoremprrngorngo 37222 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ PrRing β†’ 𝑅 ∈ RingOps)
 
Theoremsmprngopr 37223 A simple ring (one whose only ideals are 0 and 𝑅) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   ((𝑅 ∈ RingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜π‘…) = {{𝑍}, 𝑋}) β†’ 𝑅 ∈ PrRing)
 
Theoremdivrngpr 37224 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ DivRingOps β†’ 𝑅 ∈ PrRing)
 
Theoremisdmn 37225 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ Com2))
 
Theoremisdmn2 37226 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝑅 ∈ Dmn ↔ (𝑅 ∈ PrRing ∧ 𝑅 ∈ CRingOps))
 
Theoremdmncrng 37227 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn β†’ 𝑅 ∈ CRingOps)
 
Theoremdmnrngo 37228 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
(𝑅 ∈ Dmn β†’ 𝑅 ∈ RingOps)
 
Theoremflddmn 37229 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
(𝐾 ∈ Fld β†’ 𝐾 ∈ Dmn)
 
21.22.22  Ideal generators
 
Syntaxcigen 37230 Extend class notation with the ideal generation function.
class IdlGen
 
Definitiondf-igen 37231* Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
IdlGen = (π‘Ÿ ∈ RingOps, 𝑠 ∈ 𝒫 ran (1st β€˜π‘Ÿ) ↦ ∩ {𝑗 ∈ (Idlβ€˜π‘Ÿ) ∣ 𝑠 βŠ† 𝑗})
 
Theoremigenval 37232* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.) (Proof shortened by Mario Carneiro, 20-Dec-2013.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝑆 βŠ† 𝑋) β†’ (𝑅 IdlGen 𝑆) = ∩ {𝑗 ∈ (Idlβ€˜π‘…) ∣ 𝑆 βŠ† 𝑗})
 
Theoremigenss 37233 A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 βŠ† (𝑅 IdlGen 𝑆))
 
Theoremigenidl 37234 The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝑆 βŠ† 𝑋) β†’ (𝑅 IdlGen 𝑆) ∈ (Idlβ€˜π‘…))
 
Theoremigenmin 37235 The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…) ∧ 𝑆 βŠ† 𝐼) β†’ (𝑅 IdlGen 𝑆) βŠ† 𝐼)
 
Theoremigenidl2 37236 The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
((𝑅 ∈ RingOps ∧ 𝐼 ∈ (Idlβ€˜π‘…)) β†’ (𝑅 IdlGen 𝐼) = 𝐼)
 
Theoremigenval2 37237* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ RingOps ∧ 𝑆 βŠ† 𝑋) β†’ ((𝑅 IdlGen 𝑆) = 𝐼 ↔ (𝐼 ∈ (Idlβ€˜π‘…) ∧ 𝑆 βŠ† 𝐼 ∧ βˆ€π‘— ∈ (Idlβ€˜π‘…)(𝑆 βŠ† 𝑗 β†’ 𝐼 βŠ† 𝑗))))
 
Theoremprnc 37238* A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   ((𝑅 ∈ CRingOps ∧ 𝐴 ∈ 𝑋) β†’ (𝑅 IdlGen {𝐴}) = {π‘₯ ∈ 𝑋 ∣ βˆƒπ‘¦ ∈ 𝑋 π‘₯ = (𝑦𝐻𝐴)})
 
Theoremisfldidl 37239 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
𝐺 = (1st β€˜πΎ)    &   π» = (2nd β€˜πΎ)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ π‘ˆ β‰  𝑍 ∧ (Idlβ€˜πΎ) = {{𝑍}, 𝑋}))
 
Theoremisfldidl2 37240 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st β€˜πΎ)    &   π» = (2nd β€˜πΎ)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    β‡’   (𝐾 ∈ Fld ↔ (𝐾 ∈ CRingOps ∧ 𝑋 β‰  {𝑍} ∧ (Idlβ€˜πΎ) = {{𝑍}, 𝑋}))
 
Theoremispridlc 37241* The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (𝑅 ∈ CRingOps β†’ (𝑃 ∈ (PrIdlβ€˜π‘…) ↔ (𝑃 ∈ (Idlβ€˜π‘…) ∧ 𝑃 β‰  𝑋 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) ∈ 𝑃 β†’ (π‘Ž ∈ 𝑃 ∨ 𝑏 ∈ 𝑃)))))
 
Theorempridlc 37242 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ (𝐴𝐻𝐡) ∈ 𝑃)) β†’ (𝐴 ∈ 𝑃 ∨ 𝐡 ∈ 𝑃))
 
Theorempridlc2 37243 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ 𝑋 ∧ (𝐴𝐻𝐡) ∈ 𝑃)) β†’ 𝐡 ∈ 𝑃)
 
Theorempridlc3 37244 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    β‡’   (((𝑅 ∈ CRingOps ∧ 𝑃 ∈ (PrIdlβ€˜π‘…)) ∧ (𝐴 ∈ (𝑋 βˆ– 𝑃) ∧ 𝐡 ∈ (𝑋 βˆ– 𝑃))) β†’ (𝐴𝐻𝐡) ∈ (𝑋 βˆ– 𝑃))
 
Theoremisdmn3 37245* The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    &   π‘ˆ = (GIdβ€˜π»)    β‡’   (𝑅 ∈ Dmn ↔ (𝑅 ∈ CRingOps ∧ π‘ˆ β‰  𝑍 ∧ βˆ€π‘Ž ∈ 𝑋 βˆ€π‘ ∈ 𝑋 ((π‘Žπ»π‘) = 𝑍 β†’ (π‘Ž = 𝑍 ∨ 𝑏 = 𝑍))))
 
Theoremdmnnzd 37246 A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    β‡’   ((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ (𝐴𝐻𝐡) = 𝑍)) β†’ (𝐴 = 𝑍 ∨ 𝐡 = 𝑍))
 
Theoremdmncan1 37247 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    β‡’   (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ 𝐴 β‰  𝑍) β†’ ((𝐴𝐻𝐡) = (𝐴𝐻𝐢) β†’ 𝐡 = 𝐢))
 
Theoremdmncan2 37248 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
𝐺 = (1st β€˜π‘…)    &   π» = (2nd β€˜π‘…)    &   π‘‹ = ran 𝐺    &   π‘ = (GIdβ€˜πΊ)    β‡’   (((𝑅 ∈ Dmn ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋 ∧ 𝐢 ∈ 𝑋)) ∧ 𝐢 β‰  𝑍) β†’ ((𝐴𝐻𝐢) = (𝐡𝐻𝐢) β†’ 𝐴 = 𝐡))
 
21.23  Mathbox for Giovanni Mascellani
 
21.23.1  Tools for automatic proof building

The results in this section are mostly meant for being used by automatic proof building programs. As a result, they might appear less useful or meaningful than others to human beings.

 
Theoremefald2 37249 A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(Β¬ πœ‘ β†’ βŠ₯)    β‡’   πœ‘
 
Theoremnotbinot1 37250 Simplification rule of negation across a biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(Β¬ (Β¬ πœ‘ ↔ πœ“) ↔ (πœ‘ ↔ πœ“))
 
Theorembicontr 37251 Biconditional of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((Β¬ πœ‘ ↔ πœ‘) ↔ βŠ₯)
 
Theoremimpor 37252 An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((πœ‘ β†’ (πœ“ ∨ πœ’)) ↔ ((Β¬ πœ‘ ∨ πœ“) ∨ πœ’))
 
Theoremorfa 37253 The falsum βŠ₯ can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
((πœ‘ ∨ βŠ₯) ↔ πœ‘)
 
Theoremnotbinot2 37254 Commutation rule between negation and biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(Β¬ (πœ‘ ↔ πœ“) ↔ (Β¬ πœ‘ ↔ πœ“))
 
Theorembiimpor 37255 A rewriting rule for biconditional. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(((πœ‘ ↔ πœ“) β†’ πœ’) ↔ ((Β¬ πœ‘ ↔ πœ“) ∨ πœ’))
 
Theoremorfa1 37256 Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(πœ‘ β†’ πœ“)    β‡’   ((πœ‘ ∨ βŠ₯) β†’ πœ“)
 
Theoremorfa2 37257 Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(πœ‘ β†’ βŠ₯)    β‡’   ((πœ‘ ∨ πœ“) β†’ πœ“)
 
Theorembifald 37258 Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(πœ‘ β†’ Β¬ πœ“)    β‡’   (πœ‘ β†’ (πœ“ ↔ βŠ₯))
 
Theoremorsild 37259 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(πœ‘ β†’ Β¬ (πœ“ ∨ πœ’))    β‡’   (πœ‘ β†’ Β¬ πœ“)
 
Theoremorsird 37260 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
(πœ‘ β†’ Β¬ (πœ“ ∨ πœ’))    β‡’   (πœ‘ β†’ Β¬ πœ’)
 
Theoremcnf1dd 37261 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(πœ‘ β†’ (πœ“ β†’ Β¬ πœ’))    &   (πœ‘ β†’ (πœ“ β†’ (πœ’ ∨ πœƒ)))    β‡’   (πœ‘ β†’ (πœ“ β†’ πœƒ))
 
Theoremcnf2dd 37262 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(πœ‘ β†’ (πœ“ β†’ Β¬ πœƒ))    &   (πœ‘ β†’ (πœ“ β†’ (πœ’ ∨ πœƒ)))    β‡’   (πœ‘ β†’ (πœ“ β†’ πœ’))
 
Theoremcnfn1dd 37263 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(πœ‘ β†’ (πœ“ β†’ πœ’))    &   (πœ‘ β†’ (πœ“ β†’ (Β¬ πœ’ ∨ πœƒ)))    β‡’   (πœ‘ β†’ (πœ“ β†’ πœƒ))
 
Theoremcnfn2dd 37264 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(πœ‘ β†’ (πœ“ β†’ πœƒ))    &   (πœ‘ β†’ (πœ“ β†’ (πœ’ ∨ Β¬ πœƒ)))    β‡’   (πœ‘ β†’ (πœ“ β†’ πœ’))
 
Theoremor32dd 37265 A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(πœ‘ β†’ (πœ“ β†’ ((πœ’ ∨ πœƒ) ∨ 𝜏)))    β‡’   (πœ‘ β†’ (πœ“ β†’ ((πœ’ ∨ 𝜏) ∨ πœƒ)))
 
Theoremnotornotel1 37266 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(πœ‘ β†’ Β¬ (Β¬ πœ“ ∨ πœ’))    β‡’   (πœ‘ β†’ πœ“)
 
Theoremnotornotel2 37267 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(πœ‘ β†’ Β¬ (πœ“ ∨ Β¬ πœ’))    β‡’   (πœ‘ β†’ πœ’)
 
Theoremcontrd 37268 A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
(πœ‘ β†’ (Β¬ πœ“ β†’ πœ’))    &   (πœ‘ β†’ (Β¬ πœ“ β†’ Β¬ πœ’))    β‡’   (πœ‘ β†’ πœ“)
 
Theoreman12i 37269 An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.)
(πœ‘ ∧ (πœ“ ∧ πœ’))    β‡’   (πœ“ ∧ (πœ‘ ∧ πœ’))
 
Theoremexmid2 37270 An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
((πœ“ ∧ πœ‘) β†’ πœ’)    &   ((Β¬ πœ“ ∧ πœ‚) β†’ πœ’)    β‡’   ((πœ‘ ∧ πœ‚) β†’ πœ’)
 
Theoremselconj 37271 An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
(πœ‘ ↔ (πœ“ ∧ πœ’))    β‡’   ((πœ‚ ∧ πœ‘) ↔ (πœ“ ∧ (πœ‚ ∧ πœ’)))
 
Theoremtruconj 37272 Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.)
(πœ‘ ↔ (⊀ ∧ πœ‘))
 
Theoremorel 37273 An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
((πœ“ ∧ πœ‚) β†’ πœƒ)    &   ((πœ’ ∧ 𝜌) β†’ πœƒ)    &   (πœ‘ β†’ (πœ“ ∨ πœ’))    β‡’   ((πœ‘ ∧ (πœ‚ ∧ 𝜌)) β†’ πœƒ)
 
Theoremnegel 37274 An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(πœ“ β†’ πœ’)    &   (πœ‘ β†’ Β¬ πœ’)    β‡’   ((πœ‘ ∧ πœ“) β†’ βŠ₯)
 
Theorembotel 37275 An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
(πœ‘ β†’ βŠ₯)    β‡’   (πœ‘ β†’ πœ“)
 
Theoremtradd 37276 Add top ad a conjunct. (Contributed by Giovanni Mascellani, 24-May-2019.)
(πœ‘ ↔ πœ“)    β‡’   (πœ‘ ↔ (⊀ ∧ πœ“))
 
Theoremgm-sbtru 37277 Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V    β‡’   ([𝐴 / π‘₯]⊀ ↔ ⊀)
 
Theoremsbfal 37278 Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.)
𝐴 ∈ V    β‡’   ([𝐴 / π‘₯]βŠ₯ ↔ βŠ₯)
 
Theoremsbcani 37279 Distribution of class substitution over conjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
([𝐴 / π‘₯]πœ‘ ↔ πœ’)    &   ([𝐴 / π‘₯]πœ“ ↔ πœ‚)    β‡’   ([𝐴 / π‘₯](πœ‘ ∧ πœ“) ↔ (πœ’ ∧ πœ‚))
 
Theoremsbcori 37280 Distribution of class substitution over disjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
([𝐴 / π‘₯]πœ‘ ↔ πœ’)    &   ([𝐴 / π‘₯]πœ“ ↔ πœ‚)    β‡’   ([𝐴 / π‘₯](πœ‘ ∨ πœ“) ↔ (πœ’ ∨ πœ‚))
 
Theoremsbcimi 37281 Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V    &   ([𝐴 / π‘₯]πœ‘ ↔ πœ’)    &   ([𝐴 / π‘₯]πœ“ ↔ πœ‚)    β‡’   ([𝐴 / π‘₯](πœ‘ β†’ πœ“) ↔ (πœ’ β†’ πœ‚))
 
Theoremsbcni 37282 Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V    &   ([𝐴 / π‘₯]πœ‘ ↔ πœ“)    β‡’   ([𝐴 / π‘₯] Β¬ πœ‘ ↔ Β¬ πœ“)
 
Theoremsbali 37283 Discard class substitution in a universal quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V    β‡’   ([𝐴 / π‘₯]βˆ€π‘₯πœ‘ ↔ βˆ€π‘₯πœ‘)
 
Theoremsbexi 37284 Discard class substitution in an existential quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
𝐴 ∈ V    β‡’   ([𝐴 / π‘₯]βˆƒπ‘₯πœ‘ ↔ βˆƒπ‘₯πœ‘)
 
Theoremsbcalf 37285* Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
Ⅎ𝑦𝐴    β‡’   ([𝐴 / π‘₯]βˆ€π‘¦πœ‘ ↔ βˆ€π‘¦[𝐴 / π‘₯]πœ‘)
 
Theoremsbcexf 37286* Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
Ⅎ𝑦𝐴    β‡’   ([𝐴 / π‘₯]βˆƒπ‘¦πœ‘ ↔ βˆƒπ‘¦[𝐴 / π‘₯]πœ‘)
 
Theoremsbcalfi 37287* Move universal quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Ⅎ𝑦𝐴    &   ([𝐴 / π‘₯]πœ‘ ↔ πœ“)    β‡’   ([𝐴 / π‘₯]βˆ€π‘¦πœ‘ ↔ βˆ€π‘¦πœ“)
 
Theoremsbcexfi 37288* Move existential quantifier in and out of class substitution, with an explicit nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Ⅎ𝑦𝐴    &   ([𝐴 / π‘₯]πœ‘ ↔ πœ“)    β‡’   ([𝐴 / π‘₯]βˆƒπ‘¦πœ‘ ↔ βˆƒπ‘¦πœ“)
 
Theoremspsbcdi 37289 A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
𝐴 ∈ V    &   (πœ‘ β†’ βˆ€π‘₯πœ’)    &   ([𝐴 / π‘₯]πœ’ ↔ πœ“)    β‡’   (πœ‘ β†’ πœ“)
 
Theoremalrimii 37290* A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
β„²π‘¦πœ‘    &   (πœ‘ β†’ πœ“)    &   ([𝑦 / π‘₯]πœ’ ↔ πœ“)    &   β„²π‘¦πœ’    β‡’   (πœ‘ β†’ βˆ€π‘₯πœ’)
 
Theoremspesbcdi 37291 A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
(πœ‘ β†’ πœ“)    &   ([𝐴 / π‘₯]πœ’ ↔ πœ“)    β‡’   (πœ‘ β†’ βˆƒπ‘₯πœ’)
 
Theoremexlimddvf 37292 A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)
(πœ‘ β†’ βˆƒπ‘₯πœƒ)    &   β„²π‘₯πœ“    &   ((πœƒ ∧ πœ“) β†’ πœ’)    &   β„²π‘₯πœ’    β‡’   ((πœ‘ ∧ πœ“) β†’ πœ’)
 
Theoremexlimddvfi 37293 A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
(πœ‘ β†’ βˆƒπ‘₯πœƒ)    &   β„²π‘¦πœƒ    &   β„²π‘¦πœ“    &   ([𝑦 / π‘₯]πœƒ ↔ πœ‚)    &   ((πœ‚ ∧ πœ“) β†’ πœ’)    &   β„²π‘¦πœ’    β‡’   ((πœ‘ ∧ πœ“) β†’ πœ’)
 
Theoremsbceq1ddi 37294 A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
(πœ‘ β†’ 𝐴 = 𝐡)    &   (πœ“ β†’ πœƒ)    &   ([𝐴 / π‘₯]πœ’ ↔ πœƒ)    &   ([𝐡 / π‘₯]πœ’ ↔ πœ‚)    β‡’   ((πœ‘ ∧ πœ“) β†’ πœ‚)
 
Theoremsbccom2lem 37295* Lemma for sbccom2 37296. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V    β‡’   ([𝐴 / π‘₯][𝐡 / 𝑦]πœ‘ ↔ [⦋𝐴 / π‘₯⦌𝐡 / 𝑦][𝐴 / π‘₯]πœ‘)
 
Theoremsbccom2 37296* Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V    β‡’   ([𝐴 / π‘₯][𝐡 / 𝑦]πœ‘ ↔ [⦋𝐴 / π‘₯⦌𝐡 / 𝑦][𝐴 / π‘₯]πœ‘)
 
Theoremsbccom2f 37297* Commutative law for double class substitution, with nonfree variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.)
𝐴 ∈ V    &   β„²π‘¦π΄    β‡’   ([𝐴 / π‘₯][𝐡 / 𝑦]πœ‘ ↔ [⦋𝐴 / π‘₯⦌𝐡 / 𝑦][𝐴 / π‘₯]πœ‘)
 
Theoremsbccom2fi 37298* Commutative law for double class substitution, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
𝐴 ∈ V    &   β„²π‘¦π΄    &   β¦‹π΄ / π‘₯⦌𝐡 = 𝐢    &   ([𝐴 / π‘₯]πœ‘ ↔ πœ“)    β‡’   ([𝐴 / π‘₯][𝐡 / 𝑦]πœ‘ ↔ [𝐢 / 𝑦]πœ“)
 
Theoremcsbcom2fi 37299* Commutative law for double class substitution in a class, with nonfree variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
𝐴 ∈ V    &   β„²π‘¦π΄    &   β¦‹π΄ / π‘₯⦌𝐡 = 𝐢    &   β¦‹π΄ / π‘₯⦌𝐷 = 𝐸    β‡’   β¦‹π΄ / π‘₯β¦Œβ¦‹π΅ / π‘¦β¦Œπ· = ⦋𝐢 / π‘¦β¦ŒπΈ
 
21.23.2  Tseitin axioms

A collection of Tseitin axioms used to convert a wff to Conjunctive Normal Form.

 
Theoremfald 37300 Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
(πœƒ β†’ Β¬ βŠ₯)
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268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 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