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Theorem List for Metamath Proof Explorer - 36401-36500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorempm13.195 36401* Theorem *13.195 in [WhiteheadRussell] p. 179. This theorem is very similar to sbc5 3330. (Contributed by Andrew Salmon, 3-Jun-2011.) (Revised by NM, 4-Jan-2017.)
 |-  ( E. y ( y  =  A  /\  ph )  <->  [. A  /  y ]. ph )
 
Theorempm13.196a 36402* 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 36403* 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 36404* 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 36405 Equivalence between two different forms of  iota. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( E! x ph  ->  |^| { x  |  ph }  =  (
 iota x ph ) )
 
Theoremiotaexeu 36406 The iota class exists. This theorem does not require ax-nul 4556 for its proof. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( iota x ph )  e. 
 _V )
 
Theoremiotasbc 36407* 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 36408* 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 36409* 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 36410* 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 36411* 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 36412* 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 36413* 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 36414* 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 36415* 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 36416 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 36417* Theorem *14.2 in [WhiteheadRussell] p. 189. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( iota x x  =  y )  =  y
 
Theoremiotavalb 36418* Theorem *14.202 in [WhiteheadRussell] p. 189. A biconditional version of iotaval 5576. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  <->  x  =  y
 ) 
 <->  ( iota x ph )  =  y )
 )
 
Theoremiotasbc5 36419* 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 36420* 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 ) ) )
 
Theoremiotavalsb 36421* Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  x  =  y
 )  ->  ( [. y  /  z ]. ps  <->  [. ( iota x ph )  /  z ]. ps )
 )
 
Theoremsbiota1 36422 Theorem *14.25 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( A. x ( ph  ->  ps )  <->  [. ( iota x ph )  /  x ]. ps ) )
 
Theoremsbaniota 36423 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 12-Jul-2011.)
 |-  ( E! x ph  ->  ( E. x ( ph  /\  ps ) 
 <-> 
 [. ( iota x ph )  /  x ]. ps ) )
 
Theoremeubi 36424 Theorem *14.271 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  ps )  ->  ( E! x ph  <->  E! x ps )
 )
 
Theoremiotasbcq 36425 Theorem *14.272 in [WhiteheadRussell] p. 193. (Contributed by Andrew Salmon, 11-Jul-2011.)
 |-  ( A. x ( ph  <->  ps )  ->  ( [. ( iota x ph )  /  y ]. ch  <->  [. ( iota x ps )  /  y ]. ch )
 )
 
21.28.5  Set Theory
 
Theoremelnev 36426* Any set that contains one element less than the universe is not equal to it. (Contributed by Andrew Salmon, 16-Jun-2011.)
 |-  ( A  e.  _V  <->  { x  |  -.  x  =  A }  =/=  _V )
 
TheoremrusbcALT 36427 A version of Russell's paradox which is proven using proper substitution. (Contributed by Andrew Salmon, 18-Jun-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  { x  |  x  e/  x }  e/  _V
 
Theoremcompel 36428 Equivalence between two ways of saying "is a member of the complement of  A." (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( x  e.  ( _V  \  A )  <->  -.  x  e.  A )
 
Theoremcompeq 36429* Equality between two ways of saying "the complement of  A." (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( _V  \  A )  =  { x  |  -.  x  e.  A }
 
Theoremcompne 36430 The complement of  A is not equal to  A. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( _V  \  A )  =/= 
 A
 
Theoremcompab 36431 Two ways of saying "the complement of a class abstraction". (Contributed by Andrew Salmon, 15-Jul-2011.) (Proof shortened by Mario Carneiro, 11-Dec-2016.)
 |-  ( _V  \  { z  | 
 ph } )  =  { z  |  -.  ph
 }
 
Theoremconss34 36432 Contrpositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( A  C_  B  <->  ( _V  \  B )  C_  ( _V  \  A ) )
 
Theoremconss2 36433 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  ( A  C_  ( _V  \  B ) 
 <->  B  C_  ( _V  \  A ) )
 
Theoremconss1 36434 Contrapositive law for subsets. (Contributed by Andrew Salmon, 15-Jul-2011.)
 |-  (
 ( _V  \  A )  C_  B  <->  ( _V  \  B )  C_  A )
 
Theoremralbidar 36435 More general form of ralbida 2865. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( ph  ->  A. x  e.  A  ph )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  (
 A. x  e.  A  ps 
 <-> 
 A. x  e.  A  ch ) )
 
Theoremrexbidar 36436 More general form of rexbida 2941. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( ph  ->  A. x  e.  A  ph )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  ( ps  <->  ch ) )   =>    |-  ( ph  ->  ( E. x  e.  A  ps 
 <-> 
 E. x  e.  A  ch ) )
 
Theoremdropab1 36437 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  x  =  y  ->  { <. x ,  z >.  |  ph }  =  { <. y ,  z >.  |  ph } )
 
Theoremdropab2 36438 Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( A. x  x  =  y  ->  { <. z ,  x >.  |  ph }  =  { <. z ,  y >.  |  ph } )
 
Theoremipo0 36439 If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  (  _I  Po  A  <->  A  =  (/) )
 
Theoremifr0 36440 A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  (  _I  Fr  A  <->  A  =  (/) )
 
Theoremordpss 36441 ordelpss 5470 with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011.)
 |-  ( Ord  B  ->  ( A  e.  B  ->  A  C.  B ) )
 
Theoremfvsb 36442* Explicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( E! y  A F y  ->  ( [. ( F `  A )  /  x ]. ph  <->  E. x ( A. y ( A F y 
 <->  y  =  x ) 
 /\  ph ) ) )
 
Theoremfveqsb 36443* Implicit substitution of a value of a function into a wff. (Contributed by Andrew Salmon, 1-Aug-2011.)
 |-  ( x  =  ( F `  A )  ->  ( ph 
 <->  ps ) )   &    |-  F/ x ps   =>    |-  ( E! y  A F y  ->  ( ps 
 <-> 
 E. x ( A. y ( A F y 
 <->  y  =  x ) 
 /\  ph ) ) )
 
Theoremxpexb 36444 A Cartesian product exists iff its converse does. Corollary 6.9(1) in [TakeutiZaring] p. 26. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( A  X.  B )  e.  _V  <->  ( B  X.  A )  e.  _V )
 
Theoremtrelpss 36445 An element of a transitive set is a proper subset of it. Theorem 7.2 in [TakeutiZaring] p. 35. Unlike tz7.2 4838, ax-reg 8107 is required for its proof. (Contributed by Andrew Salmon, 13-Nov-2011.)
 |-  (
 ( Tr  A  /\  B  e.  A )  ->  B  C.  A )
 
21.28.6  Arithmetic
 
Theoremaddcomgi 36446 Generalization of commutative law for addition. Simplifies proofs dealing with vectors. However, it is dependent on our particular definition of ordered pair. (Contributed by Andrew Salmon, 28-Jan-2012.) (Revised by Mario Carneiro, 6-May-2015.)
 |-  ( A  +  B )  =  ( B  +  A )
 
21.28.7  Geometry
 
Syntaxcplusr 36447 Introduce the operation of vector addition.
 class  +r
 
Syntaxcminusr 36448 Introduce the operation of vector subtraction.
 class  -r
 
Syntaxctimesr 36449 Introduce the operation of scalar multiplication.
 class  .v
 
Syntaxcptdfc 36450  PtDf is a predicate that is crucial for the definition of lines as well as proving a number of important theorems.
 class  PtDf ( A ,  B )
 
Syntaxcrr3c 36451  RR3 is a class.
 class  RR3
 
Syntaxcline3 36452  line3 is a class.
 class  line3
 
Definitiondf-addr 36453* Define the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  +r  =  ( x  e.  _V ,  y  e. 
 _V  |->  ( v  e. 
 RR  |->  ( ( x `
  v )  +  ( y `  v
 ) ) ) )
 
Definitiondf-subr 36454* Define the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  -r  =  ( x  e.  _V ,  y  e. 
 _V  |->  ( v  e. 
 RR  |->  ( ( x `
  v )  -  ( y `  v
 ) ) ) )
 
Definitiondf-mulv 36455* Define the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  .v  =  ( x  e.  _V ,  y  e. 
 _V  |->  ( v  e. 
 RR  |->  ( x  x.  ( y `  v
 ) ) ) )
 
Theoremaddrval 36456* Value of the operation of vector addition. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A +r B )  =  (
 v  e.  RR  |->  ( ( A `  v
 )  +  ( B `
  v ) ) ) )
 
Theoremsubrval 36457* Value of the operation of vector subtraction. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A -r B )  =  (
 v  e.  RR  |->  ( ( A `  v
 )  -  ( B `
  v ) ) ) )
 
Theoremmulvval 36458* Value of the operation of scalar multiplication. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A .v B )  =  (
 v  e.  RR  |->  ( A  x.  ( B `
  v ) ) ) )
 
Theoremaddrfv 36459 Vector addition at a value. The operation takes each vector  A and  B and forms a new vector whose values are the sum of each of the values of  A and  B. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A +r B ) `  C )  =  ( ( A `  C )  +  ( B `  C ) ) )
 
Theoremsubrfv 36460 Vector subtraction at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A -r B ) `  C )  =  ( ( A `  C )  -  ( B `  C ) ) )
 
Theoremmulvfv 36461 Scalar multiplication at a value. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  E  /\  B  e.  D  /\  C  e.  RR )  ->  ( ( A .v B ) `  C )  =  ( A  x.  ( B `  C ) ) )
 
Theoremaddrfn 36462 Vector addition produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A +r B )  Fn  RR )
 
Theoremsubrfn 36463 Vector subtraction produces a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A -r B )  Fn  RR )
 
Theoremmulvfn 36464 Scalar multiplication producees a function. (Contributed by Andrew Salmon, 27-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A .v B )  Fn  RR )
 
Theoremaddrcom 36465 Vector addition is commutative. (Contributed by Andrew Salmon, 28-Jan-2012.)
 |-  (
 ( A  e.  C  /\  B  e.  D ) 
 ->  ( A +r B )  =  ( B +r A ) )
 
Definitiondf-ptdf 36466* Define the predicate  PtDf, which is a utility definition used to shorten definitions and simplify proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
 |-  PtDf ( A ,  B )  =  ( x  e. 
 RR  |->  ( ( ( x .v ( B -r A ) ) +v A ) " { 1 ,  2 ,  3 } ) )
 
Definitiondf-rr3 36467 Define the set of all points  RR3. We define each point  A as a function to allow the use of vector addition and subtraction as well as scalar multiplication in our proofs. (Contributed by Andrew Salmon, 15-Jul-2012.)
 |-  RR3  =  ( RR  ^m 
 { 1 ,  2 ,  3 } )
 
Definitiondf-line3 36468* Define the set of all lines. A line is an infinite subset of  RR3 that satisfies a  PtDf property. (Contributed by Andrew Salmon, 15-Jul-2012.)
 |-  line3  =  { x  e. 
 ~P RR3  |  ( 2o  ~<_  x  /\  A. y  e.  x  A. z  e.  x  (
 z  =/=  y  ->  ran  PtDf ( y ,  z
 )  =  x ) ) }
 
21.29  Mathbox for Alan Sare

We are sad to report the passing of long-time contributor Alan Sare (Nov. 9, 1954 - Mar. 23, 2019).

Alan's first contribution to Metamath was a shorter proof for tfrlem8 7110 in 2008.

He developed a tool called "completeusersproof" that assists developing proofs using his "virtual deduction" method: http://us.metamath.org/other.html#completeusersproof. His virtual deduction method is explained in the comment for wvd1 36577.

Below are some excerpts from his first emails to NM in 2007:

...I have been interested in proving set theory theorems for many years for mental exercise. I enjoy it. I have used a book by Martin Zuckerman. It is informal. I am interested in completely and perfectly proving theorems. Mr. Zuckerman leaves out most of the steps of a proof, of course, like most authors do, as you have noted. A complete proof for higher theorems would require a volume of writing similar to the metamath documents. So I am frustrated when I am not capable of constructing a proof and Zuckerman leaves out steps I do not understand. I could search for the steps in other texts, but I don't do that too much. Metamath may be the answer for me....

...If we go beyond mathematics, I believe that it is possible to write down all human knowledge in a way similar to the way you have explicated large areas of mathematics. Of course, that would be a much, much more difficult job. For example, it is possible to take a hard science like physics construct axioms based on experimental results and to cast all of physics into a collection of axioms and theorems. Maybe his has already been attempted, although I am not familiar with it. When one then moves on to the soft sciences such as social science, this job gets much more difficult. The key is: All human thought consists of logical operations on abstract objects. Usually, these logical operations are done informally. There is no reason why one cannot take any subject and explicate it and take it down to the indivisible postulates in a formal rigorous way....

...When I read a math book or an engineering book I come across something I don't understand and I am compelled to understand it. But, often it is hopeless. I don't have the time. Or, I would have to read the same thing by multiple authors in the hope that different authors would give parts of the working proof that others have omitted. It is very inefficient. Because I have always been inclined to "get to the bottom" for a 100% fully understood proof....

 
21.29.1  Auxiliary theorems for the Virtual Deduction tool
 
TheoremidiALT 36469 Placeholder for idi 2. Though unnecessary, this theorem is sometimes used in proofs in this mathbox for pedagogical purposes. (Contributed by Alan Sare, 31-Dec-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ph   =>    |-  ph
 
Theoremexbir 36470 Exportation implication also converting the consequent from a biconditional to an implication. Derived automatically from exbirVD 36889. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  (
 ( ( ph  /\  ps )  ->  ( ch  <->  th ) )  ->  ( ph  ->  ( ps  ->  ( th  ->  ch )
 ) ) )
 
Theorem3impexpbicom 36471 Version of 3impexp 1227 where in addition the consequent is commuted. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  (
 ( ( ph  /\  ps  /\ 
 ch )  ->  ( th 
 <->  ta ) )  <->  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) ) )
 
Theorem3impexpbicomi 36472 Inference associated with 3impexpbicom 36471. Derived automatically from 3impexpbicomiVD 36894. (Contributed by Alan Sare, 31-Dec-2011.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ( th  <->  ta ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ( ta  <->  th ) ) ) )
 
Theoremee02 36473 Proof of e02 36714 without virtual deductions. (Contributed by Alan Sare, 22-Jul-2012.)
 |-  ph   &    |-  ( ps  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ( th  ->  ta ) )   =>    |-  ( ps  ->  ( ch  ->  ta )
 )
 
21.29.2  Supplementary unification deductions
 
Theorembi1imp 36474 Importation inference similar to imp 430, except the outermost implication of the hypothesis is a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ch )
 )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theorembi2imp 36475 Importation inference similar to imp 430, except both implications of the hypothesis are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theorembi3impb 36476 Similar to 3impb 1201 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ( ps 
 /\  ch ) )  <->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi3impa 36477 Similar to 3impa 1200 with implication in hypothesis replaced by biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ( ph  /\  ps )  /\  ch )  <->  th )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi23impib 36478 3impib 1203 with the inner implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ( ps 
 /\  ch )  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13impib 36479 3impib 1203 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ( ps  /\  ch )  ->  th )
 )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123impib 36480 3impib 1203 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ( ps  /\  ch )  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13impia 36481 3impia 1202 with the outer implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ps )  <->  ( ch  ->  th )
 )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123impia 36482 3impia 1202 with the implications of the hypothesis biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  (
 ( ph  /\  ps )  <->  ( ch  <->  th ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi33imp12 36483 3imp 1199 with innermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  ->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi23imp13 36484 3imp 1199 with middle implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi13imp23 36485 3imp 1199 with outermost implication of the hypothesis a biconditional. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi13imp2 36486 Similar to 3imp 1199 except the outermost and innermost implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  ->  ( ch 
 <-> 
 th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi12imp3 36487 Similar to 3imp 1199 except all but innermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ( ch  ->  th ) ) )   =>    |-  ( ( ph  /\ 
 ps  /\  ch )  ->  th )
 
Theorembi23imp1 36488 Similar to 3imp 1199 except all but outermost implication are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph  ->  ( ps  <->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorembi123imp0 36489 Similar to 3imp 1199 except all implications are biconditionals. (Contributed by Alan Sare, 6-Nov-2017.)
 |-  ( ph 
 <->  ( ps  <->  ( ch  <->  th ) ) )   =>    |-  ( ( ph  /\  ps  /\ 
 ch )  ->  th )
 
Theorem4animp1 36490 A single hypothesis unification deduction with an assertion which is an implication with a 4-right-nested conjunction antecedent. (Contributed by Alan Sare, 30-May-2018.)
 |-  (
 ( ph  /\  ps  /\  ch )  ->  ( ta  <->  th ) )   =>    |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theorem4an31 36491 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
 |-  (
 ( ( ( ch 
 /\  ps )  /\  ph )  /\  th )  ->  ta )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )
 
Theorem4an4132 36492 A rearrangement of conjuncts for a 4-right-nested conjunction. (Contributed by Alan Sare, 30-May-2018.)
 |-  (
 ( ( ( th  /\ 
 ch )  /\  ps )  /\  ph )  ->  ta )   =>    |-  (
 ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  ->  ta )
 
Theoremexpcomdg 36493 Biconditional form of expcomd 439. (Contributed by Alan Sare, 22-Jul-2012.) (New usage is discouraged.)
 |-  (
 ( ph  ->  ( ( ps  /\  ch )  ->  th ) )  <->  ( ph  ->  ( ch  ->  ( ps  ->  th ) ) ) )
 
21.29.3  Conventional Metamath proofs, some derived from VD proofs
 
Theoremiidn3 36494 idn3 36632 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 23-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  ch )
 ) )
 
Theoremee222 36495 e222 36653 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 7-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( ph  ->  ( ps  ->  th )
 )   &    |-  ( ph  ->  ( ps  ->  ta ) )   &    |-  ( ch  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ps  ->  et ) )
 
Theoremee3bir 36496 Right-biconditional form of e3 36764 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 22-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ta  <->  th )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )
 
Theoremee13 36497 e13 36775 without virtual deduction connectives. Special theorem needed for the Virtual Deduction translation tool. (Contributed by Alan Sare, 28-Oct-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  ( th  ->  ta ) ) )   &    |-  ( ps  ->  ( ta  ->  et ) )   =>    |-  ( ph  ->  ( ch  ->  ( th  ->  et ) ) )
 
Theoremee121 36498 e121 36673 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ta )   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ch  ->  et ) )
 
Theoremee122 36499 e122 36670 without virtual deductions. (Contributed by Alan Sare, 13-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ps )   &    |-  ( ph  ->  ( ch  ->  th )
 )   &    |-  ( ph  ->  ( ch  ->  ta ) )   &    |-  ( ps  ->  ( th  ->  ( ta  ->  et )
 ) )   =>    |-  ( ph  ->  ( ch  ->  et ) )
 
Theoremee333 36500 e333 36760 without virtual deductions. (Contributed by Alan Sare, 17-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ph  ->  ( ps  ->  ( ch  ->  th )
 ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  ta ) ) )   &    |-  ( ph  ->  ( ps  ->  ( ch  ->  et )
 ) )   &    |-  ( th  ->  ( ta  ->  ( et  ->  ze ) ) )   =>    |-  ( ph  ->  ( ps  ->  ( ch  ->  ze )
 ) )
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