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Theorem List for Metamath Proof Explorer - 27401-27500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxctoplnd 27401 The class of Lindelöf toplogies.
 class TopLnd
 
Definitiondf-topsep 27402* A topology is separable iff it has a countable dense subset. (Contributed by Stefan O'Rear, 8-Jan-2015.)
 |- TopSep  =  {
 j  e.  Top  |  E. x  e.  ~P  U. j ( x  ~<_  om 
 /\  ( ( cls `  j ) `  x )  =  U. j ) }
 
Definitiondf-toplnd 27403* A topology is Lindelöf iff every open cover has a countable subcover. (Contributed by Stefan O'Rear, 8-Jan-2015.)
 |- TopLnd  =  { x  e.  Top  |  A. y  e.  ~P  x ( U. x  =  U. y  ->  E. z  e.  ~P  x ( z  ~<_  om 
 /\  U. x  =  U. z ) ) }
 
19.17  Mathbox for Steve Rodriguez
 
19.17.1  Miscellanea
 
Theoremiso0 27404 The empty set is an  R ,  S isomorphism from the empty set to the empty set. (Contributed by Steve Rodriguez, 24-Oct-2015.)
 |-  (/)  Isom  R ,  S  ( (/) ,  (/) )
 
Theoremssrecnpr 27405  RR is a subset of both  RR and  CC. (Contributed by Steve Rodriguez, 22-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S )
 
Theoremseff 27406 Let set  S be the reals or complexes. Then the exponential function restricted to  S is a mapping from  S to  S. (Contributed by Steve Rodriguez, 6-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   =>    |-  ( ph  ->  ( exp  |`  S ) : S --> S )
 
Theoremsblpnf 27407 The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 18380. (Contributed by Steve Rodriguez, 8-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( S  X.  S ) )   =>    |-  ( ( ph  /\  P  e.  S ) 
 ->  ( P ( ball `  D )  +oo )  =  S )
 
19.17.2  Function operations
 
Theoremcaofcan 27408* Transfer a cancellation law like mulcan 9615 to the function operation. (Contributed by Steve Rodriguez, 16-Nov-2015.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> T )   &    |-  ( ph  ->  G : A --> S )   &    |-  ( ph  ->  H : A --> S )   &    |-  ( ( ph  /\  ( x  e.  T  /\  y  e.  S  /\  z  e.  S )
 )  ->  ( ( x R y )  =  ( x R z )  <->  y  =  z
 ) )   =>    |-  ( ph  ->  (
 ( F  o F R G )  =  ( F  o F R H )  <->  G  =  H ) )
 
Theoremofsubid 27409 Function analog of subid 9277. (Contributed by Steve Rodriguez, 5-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC )  ->  ( F  o F  -  F )  =  ( A  X.  { 0 } ) )
 
Theoremofmul12 27410 Function analog of mul12 9188. (Contributed by Steve Rodriguez, 13-Nov-2015.)
 |-  (
 ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A --> CC  /\  H : A --> CC )
 )  ->  ( F  o F  x.  ( G  o F  x.  H ) )  =  ( G  o F  x.  ( F  o F  x.  H ) ) )
 
Theoremofdivrec 27411 Function analog of divrec 9650, a division analog of ofnegsub 9954. (Contributed by Steve Rodriguez, 3-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC  \  { 0 } )
 )  ->  ( F  o F  x.  (
 ( A  X.  {
 1 } )  o F  /  G ) )  =  ( F  o F  /  G ) )
 
Theoremofdivcan4 27412 Function analog of divcan4 9659. (Contributed by Steve Rodriguez, 4-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC  \  { 0 } )
 )  ->  ( ( F  o F  x.  G )  o F  /  G )  =  F )
 
Theoremofdivdiv2 27413 Function analog of divdiv2 9682. (Contributed by Steve Rodriguez, 23-Nov-2015.)
 |-  (
 ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A --> ( CC  \  { 0 } )  /\  H : A --> ( CC  \  { 0 } )
 ) )  ->  ( F  o F  /  ( G  o F  /  H ) )  =  (
 ( F  o F  x.  H )  o F  /  G ) )
 
19.17.3  Calculus
 
Theoremlhe4.4ex1a 27414 Example of the Fundamental Theorem of Calculus, part two (ftc2 19881):  S. ( 1 (,) 2 ) ( ( x ^ 2 )  -  3 )  _d x  =  -u ( 2  /  3
). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 19881 as simply the "Fundamental Theorem of Calculus", then ftc1 19879 as the "Second Fundamental Theorem of Calculus".) (Contributed by Steve Rodriguez, 28-Oct-2015.) (Revised by Steve Rodriguez, 31-Oct-2015.)
 |-  S. ( 1 (,) 2
 ) ( ( x ^ 2 )  -  3 )  _d x  =  -u ( 2  / 
 3 )
 
Theoremdvsconst 27415 Derivative of a constant function on the reals or complexes. The function may return a complex  A even if  S is  RR. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  (
 ( S  e.  { RR ,  CC }  /\  A  e.  CC )  ->  ( S  _D  ( S  X.  { A }
 ) )  =  ( S  X.  { 0 } ) )
 
Theoremdvsid 27416 Derivative of the identity function on the reals or complexes. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  (  _I  |`  S ) )  =  ( S  X.  { 1 } ) )
 
Theoremdvsef 27417 Derivative of the exponential function on the reals or complexes. (Contributed by Steve Rodriguez, 12-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  ( S  _D  ( exp  |`  S ) )  =  ( exp  |`  S ) )
 
Theoremexpgrowthi 27418* Exponential growth and decay model. See expgrowth 27420 for more information. (Contributed by Steve Rodriguez, 4-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  K  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  Y  =  ( t  e.  S  |->  ( C  x.  ( exp `  ( K  x.  t
 ) ) ) )   =>    |-  ( ph  ->  ( S  _D  Y )  =  ( ( S  X.  { K } )  o F  x.  Y ) )
 
Theoremdvconstbi 27419* The derivative of a function on  S is zero iff it is a constant function. Roughly a biconditional  S analog of dvconst 19756 and dveq0 19837. Corresponds to integration formula " S. 0  _d x  =  C " in section 4.1 of [LarsonHostetlerEdwards] p. 278. (Contributed by Steve Rodriguez, 11-Nov-2015.)
 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  Y : S --> CC )   &    |-  ( ph  ->  dom  ( S  _D  Y )  =  S )   =>    |-  ( ph  ->  ( ( S  _D  Y )  =  ( S  X.  {
 0 } )  <->  E. c  e.  CC  Y  =  ( S  X.  { c } )
 ) )
 
Theoremexpgrowth 27420* Exponential growth and decay model. The derivative of a function y of variable t equals a constant k times y itself, iff y equals some constant C times the exponential of kt. This theorem and expgrowthi 27418 illustrate one of the simplest and most crucial classes of differential equations, equations that relate functions to their derivatives.

Section 6.3 of [Strang] p. 242 calls y' = ky "the most important differential equation in applied mathematics". In the field of population ecology it is known as the Malthusian growth model or exponential law, and C, k, and t correspond to initial population size, growth rate, and time respectively (https://en.wikipedia.org/wiki/Malthusian_growth_model); and in finance, the model appears in a similar role in continuous compounding with C as the initial amount of money. In exponential decay models, k is often expressed as the negative of a positive constant λ.

Here y' is given as  ( S  _D  Y
), C as  c, and ky as  ( ( S  X.  { K }
)  o F  x.  Y ).  ( S  X.  { K } ) is the constant function that maps any real or complex input to k and  o F  x. is multiplication as a function operation.

The leftward direction of the biconditional is as given in http://www.saylor.org/site/wp-content/uploads/2011/06/MA221-2.1.1.pdf pp. 1-2, which also notes the reverse direction ("While we will not prove this here, it turns out that these are the only functions that satisfy this equation."). The rightward direction is Theorem 5.1 of [LarsonHostetlerEdwards] p. 375 (which notes " C is the initial value of y, and k is the proportionality constant. Exponential growth occurs when k > 0, and exponential decay occurs when k < 0."); its proof here closely follows the proof of y' = y in https://proofwiki.org/wiki/Exponential_Growth_Equation/Special_Case.

Statements for this and expgrowthi 27418 formulated by Mario Carneiro. (Contributed by Steve Rodriguez, 24-Nov-2015.)

 |-  ( ph  ->  S  e.  { RR ,  CC } )   &    |-  ( ph  ->  K  e.  CC )   &    |-  ( ph  ->  Y : S --> CC )   &    |-  ( ph  ->  dom  ( S  _D  Y )  =  S )   =>    |-  ( ph  ->  (
 ( S  _D  Y )  =  ( ( S  X.  { K }
 )  o F  x.  Y )  <->  E. c  e.  CC  Y  =  ( t  e.  S  |->  ( c  x.  ( exp `  ( K  x.  t ) ) ) ) ) )
 
19.18  Mathbox for Andrew Salmon
 
19.18.1  Principia Mathematica * 10
 
Theorempm10.12 27421* Theorem *10.12 in [WhiteheadRussell] p. 146. In *10, this is treated as an axiom, and the proofs in *10 are based on this theorem. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x ( ph  \/  ps )  ->  ( ph  \/  A. x ps )
 )
 
Theorempm10.14 27422 Theorem *10.14 in [WhiteheadRussell] p. 146. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( A. x ph  /\  A. x ps )  ->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps )
 )
 
Theorempm10.251 27423 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x  -.  ph  ->  -. 
 A. x ph )
 
Theorempm10.252 27424 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  E. x ph  <->  A. x  -.  ph )
 
Theorempm10.253 27425 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  A. x ph  <->  E. x  -.  ph )
 
Theoremalbitr 27426 Theorem *10.301 in [WhiteheadRussell] p. 151. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  <->  ps )  /\  A. x ( ps  <->  ch ) )  ->  A. x ( ph  <->  ch ) )
 
Theorempm10.42 27427 Theorem *10.42 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  (
 ( E. x ph  \/  E. x ps )  <->  E. x ( ph  \/  ps ) )
 
Theorempm10.52 27428* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  ->  ( A. x ( ph  ->  ps )  <->  ps ) )
 
Theorempm10.53 27429 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x ph  ->  A. x ( ph  ->  ps ) )
 
Theorempm10.541 27430* Theorem *10.541 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  \/  ps )
 ) 
 <->  ( ch  \/  A. x ( ph  ->  ps ) ) )
 
Theorempm10.542 27431* Theorem *10.542 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ch  ->  ps )
 ) 
 <->  ( ch  ->  A. x ( ph  ->  ps )
 ) )
 
Theorempm10.55 27432 Theorem *10.55 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( E. x (
 ph  /\  ps )  /\  A. x ( ph  ->  ps ) )  <->  ( E. x ph 
 /\  A. x ( ph  ->  ps ) ) )
 
Theorempm10.56 27433 Theorem *10.56 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ( ph  ->  ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x ( ps  /\  ch )
 )
 
Theorempm10.57 27434 Theorem *10.57 in [WhiteheadRussell] p. 156. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ( ph  ->  ( ps  \/  ch )
 )  ->  ( A. x ( ph  ->  ps )  \/  E. x ( ph  /\  ch )
 ) )
 
19.18.2  Principia Mathematica * 11
 
Theorem2alanimi 27435 Removes two universal quantifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ( A. x A. y ph  /\  A. x A. y ps )  ->  A. x A. y ch )
 
Theorem2al2imi 27436 Removes two universal qunatifiers from a statement. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( A. x A. y ph  ->  ( A. x A. y ps  ->  A. x A. y ch ) )
 
Theoremstdpc4-2 27437 Theorem *11.1 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ph  ->  [ z  /  x ] [ w  /  y ] ph )
 
Theorempm11.11 27438 Theorem *11.11 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ph   =>    |- 
 A. z A. w [ z  /  x ] [ w  /  y ] ph
 
Theorempm11.12 27439* Theorem *11.12 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  ->  ( ph  \/  A. x A. y ps )
 )
 
Theorem2exnaln 27440 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ph  <->  -.  A. x A. y  -.  ph )
 
Theorem2nexaln 27441 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y ph 
 <-> 
 A. x A. y  -.  ph )
 
Theorem19.21vv 27442* Compare theorem *11.3 in [WhiteheadRussell] p. 161. Special case of theorem 19.21 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  ->  ph )  <->  ( ps  ->  A. x A. y ph ) )
 
Theorem2alim 27443 Theorem *11.32 in [WhiteheadRussell] p. 162. Theorem 19.20 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( A. x A. y ph  ->  A. x A. y ps ) )
 
Theorem2albi 27444 Theorem *11.33 in [WhiteheadRussell] p. 162. Theorem 19.15 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( A. x A. y ph  <->  A. x A. y ps )
 )
 
Theorem2exim 27445 Theorem *11.34 in [WhiteheadRussell] p. 162. Theorem 19.22 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  ->  ps )  ->  ( E. x E. y ph  ->  E. x E. y ps ) )
 
Theorem2exbi 27446 Theorem *11.341 in [WhiteheadRussell] p. 162. Theorem 19.18 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph 
 <->  ps )  ->  ( E. x E. y ph  <->  E. x E. y ps )
 )
 
Theoremspsbce-2 27447 Theorem *11.36 in [WhiteheadRussell] p. 162. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  ->  E. x E. y ph )
 
Theorem19.33-2 27448 Theorem *11.421 in [WhiteheadRussell] p. 163. Theorem 19.33 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x A. y ph  \/  A. x A. y ps )  ->  A. x A. y ( ph  \/  ps ) )
 
Theorem19.36vv 27449* Theorem *11.43 in [WhiteheadRussell] p. 163. Theorem 19.36 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( E. x E. y (
 ph  ->  ps )  <->  ( A. x A. y ph  ->  ps )
 )
 
Theorem19.31vv 27450* Theorem *11.44 in [WhiteheadRussell] p. 163. Theorem 19.31 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y (
 ph  \/  ps )  <->  (
 A. x A. y ph  \/  ps ) )
 
Theorem19.37vv 27451* Theorem *11.46 in [WhiteheadRussell] p. 164. Theorem 19.37 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y ( ps  ->  ph )  <->  ( ps  ->  E. x E. y ph ) )
 
Theorem19.28vv 27452* Theorem *11.47 in [WhiteheadRussell] p. 164. Theorem 19.28 of [Margaris] p. 90 with 2 quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ps  /\  ph )  <->  ( ps  /\  A. x A. y ph ) )
 
Theorempm11.52 27453 Theorem *11.52 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y (
 ph  /\  ps )  <->  -. 
 A. x A. y
 ( ph  ->  -.  ps ) )
 
Theorem2exanali 27454 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y
 ( ph  /\  -.  ps ) 
 <-> 
 A. x A. y
 ( ph  ->  ps )
 )
 
Theoremaaanv 27455* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 1902. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ph  /\  A. y ps )  <->  A. x A. y
 ( ph  /\  ps )
 )
 
Theorempm11.57 27456* Theorem *11.57 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x ph  <->  A. x A. y
 ( ph  /\  [ y  /  x ] ph )
 )
 
Theorempm11.58 27457* Theorem *11.58 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  <->  E. x E. y
 ( ph  /\  [ y  /  x ] ph )
 )
 
Theorempm11.59 27458* Theorem *11.59 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( A. x ( ph  ->  ps )  ->  A. y A. x ( ( ph  /\ 
 [ y  /  x ] ph )  ->  ( ps  /\  [ y  /  x ] ps ) ) )
 
Theorempm11.6 27459* Theorem *11.6 in [WhiteheadRussell] p. 165. (Contributed by Andrew Salmon, 25-May-2011.)
 |-  ( E. x ( E. y
 ( ph  /\  ps )  /\  ch )  <->  E. y ( E. x ( ph  /\  ch )  /\  ps ) )
 
Theorempm11.61 27460* Theorem *11.61 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. y A. x (
 ph  ->  ps )  ->  A. x ( ph  ->  E. y ps ) )
 
Theorempm11.62 27461* Theorem *11.62 in [WhiteheadRussell] p. 166. Importation combined with the rearrangement with quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ( ph  /\  ps )  ->  ch )  <->  A. x ( ph  ->  A. y ( ps 
 ->  ch ) ) )
 
Theorempm11.63 27462 Theorem *11.63 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x E. y ph  ->  A. x A. y
 ( ph  ->  ps )
 )
 
Theorempm11.7 27463 Theorem *11.7 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x E. y (
 ph  \/  ph )  <->  E. x E. y ph )
 
Theorempm11.71 27464* Theorem *11.71 in [WhiteheadRussell] p. 166. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( E. x ph  /\ 
 E. y ch )  ->  ( ( A. x ( ph  ->  ps )  /\  A. y ( ch 
 ->  th ) )  <->  A. x A. y
 ( ( ph  /\  ch )  ->  ( ps  /\  th ) ) ) )
 
19.18.3  Predicate Calculus
 
Theoremsbeqal1 27465* If  x  =  y always implies 
x  =  z, then  y  =  z is true. (Contributed by Andrew Salmon, 2-Jun-2011.)
 |-  ( A. x ( x  =  y  ->  x  =  z )  ->  y  =  z )
 
Theoremsbeqal1i 27466* Suppose you know  x  =  y implies  x  =  z, assuming  x and  z are distinct. Then,  y  =  z. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( x  =  y  ->  x  =  z )   =>    |-  y  =  z
 
Theoremsbeqal2i 27467* If  x  =  y implies  x  =  z, then we can infer  z  =  y. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( x  =  y  ->  x  =  z )   =>    |-  z  =  y
 
Theoremsbeqalbi 27468* When both  x and  z and  y and  z are both distinct, then the converse of sbeqal1 holds as well. (Contributed by Andrew Salmon, 2-Jun-2011.)
 |-  ( x  =  y  <->  A. z ( z  =  x  ->  z  =  y ) )
 
Theoremax4567 27469 Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1552 as the inference rule. This proof extends the idea of ax467 2219 and related theorems. (Contributed by Andrew Salmon, 14-Jul-2011.)
 |-  (
 ( A. x A. y  -.  A. x A. y
 ( A. y ph  ->  ps )  ->  ( ph  ->  A. y ( A. y ph  ->  ps )
 ) )  ->  ( A. y ph  ->  A. y ps ) )
 
Theoremax4567to4 27470 Re-derivation of sp 1759 from ax4567 27469. Note that ax9 1949 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  ph )
 
Theoremax4567to5 27471 Re-derivation of ax5o 1761 from ax4567 27469. Note that only propositional calculus is required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x ( A. x ph 
 ->  ps )  ->  ( A. x ph  ->  A. x ps ) )
 
Theoremax4567to6 27472 Re-derivation of ax6o 1762 from ax4567 27469. Note that neither ax6o 1762 nor ax-7 1745 are required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax4567to7 27473 Re-derivation of ax-7 1745 from ax4567 27469. Note that ax-7 1745 is not required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Theoremax10ext 27474* This theorem shows that, given axext4 2388, we can derive a version of ax10 1991. However, it is weaker than ax10 1991 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 16-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  z  ->  A. z  z  =  x )
 
19.18.4  Principia Mathematica * 13 and * 14
 
Theorempm13.13a 27475 One result of theorem *13.13 in [WhiteheadRussell] p. 178. A note on the section - to make the theorems more usable, and because inequality is notation for set theory (it is not defined in the predicate calculus section), this section will use classes instead of sets. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  A )  ->  [. A  /  x ]. ph )
 
Theorempm13.13b 27476 Theorem *13.13 in [WhiteheadRussell] p. 178 with different variable substitution. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  x  =  A )  ->  ph )
 
Theorempm13.14 27477 Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  -.  ph )  ->  x  =/=  A )
 
Theorempm13.192 27478* Theorem *13.192 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( A. x ( x  =  A  <->  x  =  y )  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.193 27479 Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  x  =  y ) )
 
Theorempm13.194 27480 Theorem *13.194 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  ph  /\  x  =  y ) )
 
Theorempm13.195 27481* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3145. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( y  =  A  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.196a 27482* Theorem *13.196 in [WhiteheadRussell] p. 179. The only difference is the position of the substituted variable. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  ( -.  ph  <->  A. y ( [
 y  /  x ] ph  ->  y  =/=  x ) )
 
Theorem2sbc6g 27483* Theorem *13.21 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A. z A. w ( ( z  =  A  /\  w  =  B )  ->  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theorem2sbc5g 27484* Theorem *13.22 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( E. z E. w ( ( z  =  A  /\  w  =  B )  /\  ph )  <->  [. A  /  z ]. [. B  /  w ]. ph ) )
 
Theoremiotain 27485 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( E! x ph  ->  |^| { x  |  ph }  =  (
 iota x ph ) )
 
Theoremiotaexeu 27486 The iota class exists. This theorem does not require ax-nul 4298 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  e. 
 _V )
 
Theoremiotasbc 27487* Definition *14.01 in [WhiteheadRussell] p. 184. In Principia Mathematica, Russell and Whitehead define  iota in terms of a function of  ( iota x ph ). Their definition differs in that a function of  ( iota x ph ) evaluates to "false" when there isn't a single  x that satisfies  ph. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( A. x ( ph  <->  x  =  y
 )  /\  ps )
 ) )
 
Theoremiotasbc2 27488* Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  (
 ( E! x ph  /\ 
 E! x ps )  ->  ( [. ( iota
 x ph )  /  y ]. [. ( iota x ps )  /  z ]. ch  <->  E. y E. z
 ( A. x ( ph  <->  x  =  y )  /\  A. x ( ps  <->  x  =  z
 )  /\  ch )
 ) )
 
Theorempm14.12 27489* Theorem *14.12 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  A. x A. y ( ( ph  /\  [. y  /  x ].
 ph )  ->  x  =  y ) )
 
Theorempm14.122a 27490* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  [. A  /  x ]. ph )
 ) )
 
Theorempm14.122b 27491* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  ( ( A. x (
 ph  ->  x  =  A )  /\  [. A  /  x ].
 ph )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.122c 27492* Theorem *14.122 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  ( A  e.  V  ->  (
 A. x ( ph  <->  x  =  A )  <->  ( A. x ( ph  ->  x  =  A )  /\  E. x ph ) ) )
 
Theorempm14.123a 27493* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph ) ) )
 
Theorempm14.123b 27494* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  [. A  /  z ]. [. B  /  w ]. ph )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.123c 27495* Theorem *14.123 in [WhiteheadRussell] p. 185. (Contributed by Andrew Salmon, 9-Jun-2011.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( A. z A. w ( ph  <->  ( z  =  A  /\  w  =  B ) )  <->  ( A. z A. w ( ph  ->  ( z  =  A  /\  w  =  B )
 )  /\  E. z E. w ph ) ) )
 
Theorempm14.18 27496 Theorem *14.18 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ps  ->  [. ( iota x ph )  /  x ]. ps ) )
 
Theoremiotaequ 27497* Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( iota x x  =  y )  =  y
 
Theoremiotavalb 27498* Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5388. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y
 ) 
 <->  ( iota x ph )  =  y )
 )
 
Theoremiotasbc5 27499* Theorem *14.205 in [WhiteheadRussell] p. 190. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( [. ( iota x ph )  /  y ]. ps  <->  E. y ( y  =  ( iota x ph )  /\  ps ) ) )
 
Theorempm14.24 27500* Theorem *14.24 in [WhiteheadRussell] p. 191. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  A. y
 ( [. y  /  x ].
 ph 
 <->  y  =  ( iota
 x ph ) ) )
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