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Theorem List for Metamath Proof Explorer - 26901-27000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxctoplnd 26901 The class of Lindelöf toplogies.
 class TopLnd
 
Definitiondf-topsep 26902* 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 26903* 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 ) ) }
 
18.17  Mathbox for Steve Rodriguez
 
18.17.1  Miscellanea
 
Theoremiso0 26904 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 26905  RR is a subset of both  RR and  CC. (Contributed by Steve Rodriguez, 22-Nov-2015.)
 |-  ( S  e.  { RR ,  CC }  ->  RR  C_  S )
 
Theoremseff 26906 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 26907 The infinity ball in the absolute value metric is just the whole space.  S analog of blpnf 17917. (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 )
 
18.17.2  Function operations
 
Theoremcaofcan 26908* Transfer a cancellation law like mulcan 9373 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 26909 Function analog of subid 9035. (Contributed by Steve Rodriguez, 5-Nov-2015.)
 |-  (
 ( A  e.  V  /\  F : A --> CC )  ->  ( F  o F  -  F )  =  ( A  X.  { 0 } ) )
 
Theoremofmul12 26910 Function analog of mul12 8946. (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 26911 Function analog of divrec 9408, a division analog of ofnegsub 9712. (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 26912 Function analog of divcan4 9417. (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 26913 Function analog of divdiv2 9440. (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 ) )
 
18.17.3  Calculus
 
Theoremlhe4.4ex1a 26914 Example of the Fundamental Theorem of Calculus, part two (ftc2 19354):  S. ( 1 (,) 2 ) ( ( x ^ 2 )  -  3 )  _d x  =  -u ( 2  /  3
). Section 4.4 example 1a of [LarsonHostetlerEdwards] p. 311. (The book teaches ftc2 19354 as simply the "Fundamental Theorem of Calculus", then ftc1 19352 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 26915 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 26916 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 26917 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 26918* Exponential growth and decay model. See expgrowth 26920 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 26919* The derivative of a function on  S is zero iff it is a constant function. Roughly a biconditional  S analog of dvconst 19229 and dveq0 19310. 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 26920* 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 26918 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 26918 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 ) ) ) ) ) )
 
18.18  Mathbox for Andrew Salmon
 
18.18.1  Principia Mathematica * 10
 
Theorempm10.12 26921* 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 26922 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 26923 Theorem *10.251 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( A. x  -.  ph  ->  -. 
 A. x ph )
 
Theorempm10.252 26924 Theorem *10.252 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  E. x ph  <->  A. x  -.  ph )
 
Theorempm10.253 26925 Theorem *10.253 in [WhiteheadRussell] p. 149. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( -.  A. x ph  <->  E. x  -.  ph )
 
Theoremalbitr 26926 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 26927 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 26928* Theorem *10.52 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( E. x ph  ->  ( A. x ( ph  ->  ps )  <->  ps ) )
 
Theorempm10.53 26929 Theorem *10.53 in [WhiteheadRussell] p. 155. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( -.  E. x ph  ->  A. x ( ph  ->  ps ) )
 
Theorempm10.541 26930* 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 26931* 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 26932 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 26933 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 26934 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 )
 ) )
 
18.18.2  Principia Mathematica * 11
 
Theorem2alanimi 26935 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 26936 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 26937 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 26938 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 26939* 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 26940 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 26941 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 26942* 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 26943 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 26944 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 26945 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 26946 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 )
 )
 
Theorema4sbce-2 26947 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 26948 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 26949* 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 26950* 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 26951* 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 26952* 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 26953 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 26954 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 26955* Theorem *11.56 in [WhiteheadRussell] p. 165. Special case of aaan 1811. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  (
 ( A. x ph  /\  A. y ps )  <->  A. x A. y
 ( ph  /\  ps )
 )
 
Theorempm11.57 26956* 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 26957* 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 26958* 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 26959* 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 26960* 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 26961* 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 26962 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 26963 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 26964* 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 ) ) ) )
 
18.18.3  Predicate Calculus
 
Theoremsbeqal1 26965* 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 26966* 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 26967* 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 26968* 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 26969 Proof of a theorem that can act as a sole axiom for pure predicate calculus with ax-gen 1536 as the inference rule. This proof extends the idea of ax467 1752 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 26970 Re-derivation of ax-4 1692 from ax4567 26969. Note that ax-9 1684 is used for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  ph )
 
Theoremax4567to5 26971 Re-derivation of ax-5o 1694 from ax4567 26969. 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 26972 Re-derivation of ax-6o 1697 from ax4567 26969. Note that neither ax-6o 1697 nor ax-7 1535 are required for the re-derivation. (Contributed by Andrew Salmon, 14-Jul-2011.) (Proof modification is discouraged.)
 |-  ( -.  A. x  -.  A. x ph  ->  ph )
 
Theoremax4567to7 26973 Re-derivation of ax-7 1535 from ax4567 26969. Note that ax-7 1535 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 26974* This theorem shows that, given axext4 2242, we can derive a version of ax-10 1678. However, it is weaker than ax-10 1678 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 )
 
Theoremax10-16 26975* This theorem shows that, given ax-16 1927, we can derive a version of ax-10 1678. However, it is weaker than ax-10 1678 because it has a distinct variable requirement. (Contributed by Andrew Salmon, 27-Jul-2011.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  z  ->  A. z  z  =  x )
 
18.18.4  Principia Mathematica * 13 and * 14
 
Theorempm13.13a 26976 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 26977 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 26978 Theorem *13.14 in [WhiteheadRussell] p. 178. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( [. A  /  x ].
 ph  /\  -.  ph )  ->  x  =/=  A )
 
Theorempm13.192 26979* 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 26980 Theorem *13.193 in [WhiteheadRussell] p. 179. (Contributed by Andrew Salmon, 3-Jun-2011.)
 |-  (
 ( ph  /\  x  =  y )  <->  ( [ y  /  x ] ph  /\  x  =  y ) )
 
Theorempm13.194 26981 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 26982* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 2990. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( y  =  A  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.196a 26983* 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 26984* 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 26985* 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 26986 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( E! x ph  ->  |^| { x  |  ph }  =  (
 iota x ph ) )
 
Theoremiotaexeu 26987 The iota class exists. This theorem does not require ax-nul 4123 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  e. 
 _V )
 
Theoremiotasbc 26988* 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 26989* 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 26990* 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 26991* 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 26992* 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 26993* 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 26994* 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 26995* 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 26996* 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 26997 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 26998* Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( iota x x  =  y )  =  y
 
Theoremiotavalb 26999* Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 6236. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y
 ) 
 <->  ( iota x ph )  =  y )
 )
 
Theoremiotasbc5 27000* 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 ) ) )
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