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Theorem List for Metamath Proof Explorer - 32201-32300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-prrngo 32201 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  =  {
 r  e.  RingOps  |  {
 (GId `  ( 1st `  r ) ) }  e.  ( PrIdl `  r ) }
 
Definitiondf-dmn 32202 Define the class of (integral) domains. A domain is a commutative prime ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  Dmn  =  ( PrRing  i^i  Com2 )
 
Theoremisprrngo 32203 The predicate "is a prime ring". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   =>    |-  ( R  e.  PrRing  <->  ( R  e.  RingOps  /\  { Z }  e.  ( PrIdl `  R ) ) )
 
Theoremprrngorngo 32204 A prime ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e.  PrRing  ->  R  e. 
 RingOps )
 
Theoremsmprngopr 32205 A simple ring (one whose only ideals are  0 and  R) is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  U  =/=  Z  /\  ( Idl `  R )  =  { { Z } ,  X } )  ->  R  e.  PrRing )
 
Theoremdivrngpr 32206 A division ring is a prime ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( R  e.  DivRingOps  ->  R  e.  PrRing )
 
Theoremisdmn 32207 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e.  Com2 )
 )
 
Theoremisdmn2 32208 The predicate "is a domain". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e.  Dmn  <->  ( R  e.  PrRing  /\  R  e. CRingOps ) )
 
Theoremdmncrng 32209 A domain is a commutative ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( R  e.  Dmn  ->  R  e. CRingOps )
 
Theoremdmnrngo 32210 A domain is a ring. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  ( R  e.  Dmn  ->  R  e. 
 RingOps )
 
Theoremflddmn 32211 A field is a domain. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( K  e.  Fld  ->  K  e.  Dmn )
 
21.18.19  Ideal generators
 
Syntaxcigen 32212 Extend class notation with the ideal generation function.
 class  IdlGen
 
Definitiondf-igen 32213* Define the ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  IdlGen  =  ( r  e.  RingOps ,  s  e.  ~P ran  ( 1st `  r )  |->  |^| { j  e.  ( Idl `  r
 )  |  s  C_  j } )
 
Theoremigenval 32214* 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 `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  =  |^| { j  e.  ( Idl `  R )  |  S  C_  j } )
 
Theoremigenss 32215 A set is a subset of the ideal it generates. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  S  C_  ( R  IdlGen  S ) )
 
Theoremigenidl 32216 The ideal generated by a set is an ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  ( R  IdlGen  S )  e.  ( Idl `  R ) )
 
Theoremigenmin 32217 The ideal generated by a set is the minimal ideal containing that set. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R )  /\  S  C_  I )  ->  ( R  IdlGen  S )  C_  I )
 
Theoremigenidl2 32218 The ideal generated by an ideal is that ideal. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  (
 ( R  e.  RingOps  /\  I  e.  ( Idl `  R ) )  ->  ( R  IdlGen  I )  =  I )
 
Theoremigenval2 32219* The ideal generated by a subset of a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   =>    |-  (
 ( R  e.  RingOps  /\  S  C_  X )  ->  ( ( R  IdlGen  S )  =  I  <->  ( I  e.  ( Idl `  R )  /\  S  C_  I  /\  A. j  e.  ( Idl `  R ) ( S  C_  j  ->  I 
 C_  j ) ) ) )
 
Theoremprnc 32220* A principal ideal (an ideal generated by one element) in a commutative ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  A  e.  X )  ->  ( R  IdlGen  { A } )  =  { x  e.  X  |  E. y  e.  X  x  =  ( y H A ) } )
 
Theoremisfldidl 32221 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  K )   &    |-  H  =  ( 2nd `  K )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( K  e.  Fld  <->  ( K  e. CRingOps  /\  U  =/=  Z 
 /\  ( Idl `  K )  =  { { Z } ,  X }
 ) )
 
Theoremisfldidl2 32222 Determine if a ring is a field based on its ideals. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  K )   &    |-  H  =  ( 2nd `  K )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( K  e.  Fld  <->  ( K  e. CRingOps  /\  X  =/=  { Z }  /\  ( Idl `  K )  =  { { Z } ,  X } ) )
 
Theoremispridlc 32223* The predicate "is a prime ideal". Alternate definition for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e. CRingOps  ->  ( P  e.  ( PrIdl `  R )  <->  ( P  e.  ( Idl `  R )  /\  P  =/=  X  /\  A. a  e.  X  A. b  e.  X  (
 ( a H b )  e.  P  ->  ( a  e.  P  \/  b  e.  P )
 ) ) ) )
 
Theorempridlc 32224 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R ) ) 
 /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  e.  P )
 )  ->  ( A  e.  P  \/  B  e.  P ) )
 
Theorempridlc2 32225 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R ) ) 
 /\  ( A  e.  ( X  \  P ) 
 /\  B  e.  X  /\  ( A H B )  e.  P )
 )  ->  B  e.  P )
 
Theorempridlc3 32226 Property of a prime ideal in a commutative ring. (Contributed by Jeff Madsen, 17-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( ( R  e. CRingOps  /\  P  e.  ( PrIdl `  R ) ) 
 /\  ( A  e.  ( X  \  P ) 
 /\  B  e.  ( X  \  P ) ) )  ->  ( A H B )  e.  ( X  \  P ) )
 
Theoremisdmn3 32227* The predicate "is a domain", alternate expression. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  Dmn  <->  ( R  e. CRingOps  /\  U  =/=  Z 
 /\  A. a  e.  X  A. b  e.  X  ( ( a H b )  =  Z  ->  ( a  =  Z  \/  b  =  Z )
 ) ) )
 
Theoremdmnnzd 32228 A domain has no zero-divisors (besides zero). (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  ( A H B )  =  Z ) ) 
 ->  ( A  =  Z  \/  B  =  Z ) )
 
Theoremdmncan1 32229 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  /\  A  =/=  Z )  ->  ( ( A H B )  =  ( A H C )  ->  B  =  C )
 )
 
Theoremdmncan2 32230 Cancellation law for domains. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  Z  =  (GId `  G )   =>    |-  ( ( ( R  e.  Dmn  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) )  /\  C  =/=  Z )  ->  ( ( A H C )  =  ( B H C )  ->  A  =  B )
 )
 
21.19  Mathbox for Giovanni Mascellani
 
21.19.1  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 32231 A proof by contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( -.  ph  -> F.  )   =>    |-  ph
 
Theoremnotbinot1 32232 Simplification rule of negation across a biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( -.  ( -.  ph  <->  ps )  <->  ( ph  <->  ps ) )
 
Theorembicontr 32233 Biimplication of its own negation is a contradiction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  (
 ( -.  ph  <->  ph )  <-> F.  )
 
Theoremimpor 32234 An equivalent formula for implying a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  (
 ( ph  ->  ( ps 
 \/  ch ) )  <->  ( ( -.  ph  \/  ps )  \/ 
 ch ) )
 
Theoremorfa 32235 The falsum F. can be removed from a disjunction. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  (
 ( ph  \/ F.  )  <->  ph )
 
Theoremnotbinot2 32236 Commutation rule between negation and biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( -.  ( ph  <->  ps )  <->  ( -.  ph  <->  ps ) )
 
Theorembiimpor 32237 A rewriting rule for biimplication. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  (
 ( ( ph  <->  ps )  ->  ch )  <->  ( ( -.  ph  <->  ps )  \/  ch ) )
 
Theoremunitresl 32238 A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  -. 
 ch )   =>    |-  ( ph  ->  ps )
 
Theoremunitresr 32239 A lemma for Conjunctive Normal Form unit propagation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( ph  ->  ( ps  \/  ch ) )   &    |-  ( ph  ->  -. 
 ps )   =>    |-  ( ph  ->  ch )
 
Theoremorfa1 32240 Add a contradicting disjunct to an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( ph  ->  ps )   =>    |-  ( ( ph  \/ F.  )  ->  ps )
 
Theoremorfa2 32241 Remove a contradicting disjunct from an antecedent. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( ph  -> F.  )   =>    |-  ( ( ph  \/  ps )  ->  ps )
 
Theorembifald 32242 Infer the equivalence to a contradiction from a negation, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  ( ps  <-> F.  ) )
 
Theoremorsild 32243 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( ph  ->  -.  ( ps  \/  ch ) )   =>    |-  ( ph  ->  -. 
 ps )
 
Theoremorsird 32244 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 15-Sep-2017.)
 |-  ( ph  ->  -.  ( ps  \/  ch ) )   =>    |-  ( ph  ->  -. 
 ch )
 
Theoremorcomdd 32245 Commutativity of logic disjunction, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
 |-  ( ph  ->  ( ps  ->  ( ch  \/  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ( th  \/  ch ) ) )
 
Theoremcnf1dd 32246 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
 |-  ( ph  ->  ( ps  ->  -. 
 ch ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  \/  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremcnf2dd 32247 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
 |-  ( ph  ->  ( ps  ->  -. 
 th ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  \/  th )
 ) )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremcnfn1dd 32248 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  ( -.  ch 
 \/  th ) ) )   =>    |-  ( ph  ->  ( ps  ->  th ) )
 
Theoremcnfn2dd 32249 A lemma for Conjunctive Normal Form unit propagation, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
 |-  ( ph  ->  ( ps  ->  th ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  \/  -.  th ) ) )   =>    |-  ( ph  ->  ( ps  ->  ch ) )
 
Theoremor32dd 32250 A rearrangement of disjuncts, in double deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
 |-  ( ph  ->  ( ps  ->  ( ( ch  \/  th )  \/  ta ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ( ch 
 \/  ta )  \/  th ) ) )
 
Theoremnotornotel1 32251 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
 |-  ( ph  ->  -.  ( -.  ps 
 \/  ch ) )   =>    |-  ( ph  ->  ps )
 
Theoremnotornotel2 32252 A lemma for not-or-not elimination, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
 |-  ( ph  ->  -.  ( ps  \/  -.  ch ) )   =>    |-  ( ph  ->  ch )
 
Theoremcontrd 32253 A proof by contradiction, in deduction form. (Contributed by Giovanni Mascellani, 19-Mar-2018.)
 |-  ( ph  ->  ( -.  ps  ->  ch ) )   &    |-  ( ph  ->  ( -.  ps  ->  -.  ch ) )   =>    |-  ( ph  ->  ps )
 
Theoreman12i 32254 An inference from commuting operands in a chain of conjunctions. (Contributed by Giovanni Mascellani, 22-May-2019.)
 |-  ( ph  /\  ( ps  /\  ch ) )   =>    |-  ( ps  /\  ( ph  /\  ch ) )
 
Theoremexmid2 32255 An excluded middle law. (Contributed by Giovanni Mascellani, 23-May-2019.)
 |-  (
 ( ps  /\  ph )  ->  ch )   &    |-  ( ( -. 
 ps  /\  et )  ->  ch )   =>    |-  ( ( ph  /\  et )  ->  ch )
 
Theoremselconj 32256 An inference for selecting one of a list of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
 |-  ( ph 
 <->  ( ps  /\  ch ) )   =>    |-  ( ( et  /\  ph )  <->  ( ps  /\  ( et  /\  ch )
 ) )
 
Theoremtruconj 32257 Add true as a conjunct. (Contributed by Giovanni Mascellani, 23-May-2019.)
 |-  ( ph 
 <->  ( T.  /\  ph )
 )
 
Theoremmergeconj 32258 An inference to merge two lists of conjuncts. (Contributed by Giovanni Mascellani, 23-May-2019.)
 |-  ( ph 
 <->  ( ps  /\  ch ) )   =>    |-  ( ( et  /\  ph )  <->  ( ( et 
 /\  ps )  /\  ch ) )
 
Theoremorel 32259 An inference for disjunction elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
 |-  (
 ( ps  /\  et )  ->  th )   &    |-  ( ( ch 
 /\  rh )  ->  th )   &    |-  ( ph  ->  ( ps  \/  ch ) )   =>    |-  ( ( ph  /\  ( et  /\  rh ) ) 
 ->  th )
 
Theoremnegel 32260 An inference for negation elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
 |-  ( ps  ->  ch )   &    |-  ( ph  ->  -. 
 ch )   =>    |-  ( ( ph  /\  ps )  -> F.  )
 
Theorembotel 32261 An inference for bottom elimination. (Contributed by Giovanni Mascellani, 24-May-2019.)
 |-  ( ph  -> F.  )   =>    |-  ( ph  ->  ps )
 
Theoremtradd 32262 Add top ad a conjunct. (Contributed by Giovanni Mascellani, 24-May-2019.)
 |-  ( ph 
 <->  ps )   =>    |-  ( ph  <->  ( T.  /\  ps ) )
 
Theoremsbtru 32263 Substitution does not change truth. (Contributed by Giovanni Mascellani, 24-May-2019.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. T.  <-> T.  )
 
Theoremsbfal 32264 Substitution does not change falsity. (Contributed by Giovanni Mascellani, 24-May-2019.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. F.  <-> F.  )
 
Theoremsbcani 32265 Distribution of class substitution over conjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
 |-  ( [. A  /  x ].
 ph 
 <->  ch )   &    |-  ( [. A  /  x ]. ps  <->  et )   =>    |-  ( [. A  /  x ]. ( ph  /\  ps ) 
 <->  ( ch  /\  et ) )
 
Theoremsbcori 32266 Distribution of class substitution over disjunction, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
 |-  ( [. A  /  x ].
 ph 
 <->  ch )   &    |-  ( [. A  /  x ]. ps  <->  et )   =>    |-  ( [. A  /  x ]. ( ph  \/  ps )  <->  ( ch  \/  et ) )
 
Theoremsbcimi 32267 Distribution of class substitution over implication, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
 |-  A  e.  _V   &    |-  ( [. A  /  x ]. ph  <->  ch )   &    |-  ( [. A  /  x ]. ps  <->  et )   =>    |-  ( [. A  /  x ]. ( ph  ->  ps )  <->  ( ch  ->  et ) )
 
Theoremsbceqi 32268 Distribution of class substitution over equality, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
 |-  A  e.  _V   &    |-  [_ A  /  x ]_ B  =  D   &    |-  [_ A  /  x ]_ C  =  E   =>    |-  ( [. A  /  x ]. B  =  C  <->  D  =  E )
 
Theoremsbcni 32269 Move class substitution inside a negation, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
 |-  A  e.  _V   &    |-  ( [. A  /  x ]. ph  <->  ps )   =>    |-  ( [. A  /  x ].  -.  ph  <->  -.  ps )
 
Theoremsbali 32270 Discard class substitution in a universal quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. A. x ph  <->  A. x ph )
 
Theoremsbexi 32271 Discard class substitution in an existential quantification when substituting the quantified variable, in inference form. (Contributed by Giovanni Mascellani, 27-May-2019.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. E. x ph  <->  E. x ph )
 
Theoremsbcalf 32272* Move universal quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
 |-  F/_ y A   =>    |-  ( [. A  /  x ]. A. y ph  <->  A. y [. A  /  x ].
 ph )
 
Theoremsbcexf 32273* Move existential quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
 |-  F/_ y A   =>    |-  ( [. A  /  x ]. E. y ph  <->  E. y [. A  /  x ].
 ph )
 
Theoremsbcalfi 32274* Move universal quantifier in and out of class substitution, with an explicit non-free variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
 |-  F/_ y A   &    |-  ( [. A  /  x ]. ph  <->  ps )   =>    |-  ( [. A  /  x ]. A. y ph  <->  A. y ps )
 
Theoremsbcexfi 32275* Move existential quantifier in and out of class substitution, with an explicit non-free variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
 |-  F/_ y A   &    |-  ( [. A  /  x ]. ph  <->  ps )   =>    |-  ( [. A  /  x ]. E. y ph  <->  E. y ps )
 
Theoremcsbvargi 32276 The proper substitution of a class for a variable in that variable results in the class (if the class exists), in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
 |-  A  e.  _V   =>    |-  [_ A  /  x ]_ x  =  A
 
Theoremcsbconstgi 32277* The proper substitution of a class for a variable in another variable does not modify it, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
 |-  A  e.  _V   =>    |-  [_ A  /  x ]_ y  =  y
 
Theoremspsbcdi 32278 A lemma for eliminating a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
 |-  A  e.  _V   &    |-  ( ph  ->  A. x ch )   &    |-  ( [. A  /  x ].
 ch 
 <->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremalrimii 32279* A lemma for introducing a universal quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
 |-  F/ y ph   &    |-  ( ph  ->  ps )   &    |-  ( [. y  /  x ]. ch  <->  ps )   &    |-  F/ y ch   =>    |-  ( ph  ->  A. x ch )
 
Theoremspesbcdi 32280 A lemma for introducing an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
 |-  ( ph  ->  ps )   &    |-  ( [. A  /  x ]. ch  <->  ps )   =>    |-  ( ph  ->  E. x ch )
 
Theoremexlimddvf 32281 A lemma for eliminating an existential quantifier. (Contributed by Giovanni Mascellani, 30-May-2019.)
 |-  ( ph  ->  E. x th )   &    |-  F/ x ps   &    |-  ( ( th  /\ 
 ps )  ->  ch )   &    |-  F/ x ch   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremexlimddvfi 32282 A lemma for eliminating an existential quantifier, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
 |-  ( ph  ->  E. x th )   &    |-  F/ y th   &    |-  F/ y ps   &    |-  ( [. y  /  x ].
 th 
 <->  et )   &    |-  ( ( et 
 /\  ps )  ->  ch )   &    |-  F/ y ch   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremsbceq1ddi 32283 A lemma for eliminating inequality, in inference form. (Contributed by Giovanni Mascellani, 31-May-2019.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ps  ->  th )   &    |-  ( [. A  /  x ].
 ch 
 <-> 
 th )   &    |-  ( [. B  /  x ]. ch  <->  et )   =>    |-  ( ( ph  /\  ps )  ->  et )
 
Theoremsbccom2lem 32284* Lemma for sbccom2 32285. (Contributed by Giovanni Mascellani, 31-May-2019.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbccom2 32285* Commutative law for double class substitution. (Contributed by Giovanni Mascellani, 31-May-2019.)
 |-  A  e.  _V   =>    |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbccom2f 32286* Commutative law for double class substitution, with non free variable condition. (Contributed by Giovanni Mascellani, 31-May-2019.)
 |-  A  e.  _V   &    |-  F/_ y A   =>    |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. [_ A  /  x ]_ B  /  y ]. [. A  /  x ].
 ph )
 
Theoremsbccom2fi 32287* Commutative law for double class substitution, with non free variable condition and in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
 |-  A  e.  _V   &    |-  F/_ y A   &    |-  [_ A  /  x ]_ B  =  C   &    |-  ( [. A  /  x ]. ph  <->  ps )   =>    |-  ( [. A  /  x ]. [. B  /  y ]. ph  <->  [. C  /  y ]. ps )
 
Theoremsbcgfi 32288 Substitution for a variable not free in a wff does not affect it, in inference form. (Contributed by Giovanni Mascellani, 1-Jun-2019.)
 |-  A  e.  _V   &    |-  F/ x ph   =>    |-  ( [. A  /  x ].
 ph 
 <-> 
 ph )
 
Theoremcsbcom2fi 32289* Commutative law for double class substitution in a class, with non free variable condition and in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
 |-  A  e.  _V   &    |-  F/_ y A   &    |-  [_ A  /  x ]_ B  =  C   &    |-  [_ A  /  x ]_ D  =  E   =>    |-  [_ A  /  x ]_
 [_ B  /  y ]_ D  =  [_ C  /  y ]_ E
 
Theoremcsbgfi 32290 Substitution for a variable not free in a class does not affect it, in inference form. (Contributed by Giovanni Mascellani, 4-Jun-2019.)
 |-  A  e.  _V   &    |-  F/_ x B   =>    |-  [_ A  /  x ]_ B  =  B
 
21.19.2  Tseitin axioms

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

 
Theoremfald 32291 Refutation of falsity, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  -. F.  )
 
Theoremtsim1 32292 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  ( ( -.  ph  \/  ps )  \/ 
 -.  ( ph  ->  ps ) ) )
 
Theoremtsim2 32293 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  ( ph  \/  ( ph  ->  ps )
 ) )
 
Theoremtsim3 32294 A Tseitin axiom for logical implication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  ( -.  ps  \/  ( ph  ->  ps )
 ) )
 
Theoremtsbi1 32295 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  ( ( -.  ph  \/  -.  ps )  \/  ( ph  <->  ps ) ) )
 
Theoremtsbi2 32296 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  ( ( ph  \/  ps )  \/  ( ph 
 <->  ps ) ) )
 
Theoremtsbi3 32297 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  ( ( ph  \/  -.  ps )  \/ 
 -.  ( ph  <->  ps ) ) )
 
Theoremtsbi4 32298 A Tseitin axiom for logical biimplication, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  ( ( -.  ph  \/  ps )  \/ 
 -.  ( ph  <->  ps ) ) )
 
Theoremtsxo1 32299 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  ( ( -.  ph  \/  -.  ps )  \/  -.  ( ph  \/_  ps ) ) )
 
Theoremtsxo2 32300 A Tseitin axiom for logical exclusive disjunction, in deduction form. (Contributed by Giovanni Mascellani, 24-Mar-2018.)
 |-  ( th  ->  ( ( ph  \/  ps )  \/  -.  ( ph  \/_  ps )
 ) )
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