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Theorem List for Metamath Proof Explorer - 3901-4000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrzal 3901* Vacuous quantification is always true. (Contributed by NM, 11-Mar-1997.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( A  =  (/)  ->  A. x  e.  A  ph )
 
Theoremrexn0 3902* Restricted existential quantification implies its restriction is nonempty. (Contributed by Szymon Jaroszewicz, 3-Apr-2007.)
 |-  ( E. x  e.  A  ph  ->  A  =/=  (/) )
 
Theoremralidm 3903* Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
 |-  ( A. x  e.  A  A. x  e.  A  ph  <->  A. x  e.  A  ph )
 
Theoremral0 3904 Vacuous universal quantification is always true. (Contributed by NM, 20-Oct-2005.)
 |- 
 A. x  e.  (/)  ph
 
Theoremrgenz 3905* Generalization rule that eliminates a non-zero class requirement. (Contributed by NM, 8-Dec-2012.)
 |-  ( ( A  =/=  (/)  /\  x  e.  A )  ->  ph )   =>    |- 
 A. x  e.  A  ph
 
Theoremralf0 3906* The quantification of a falsehood is vacuous when true. (Contributed by NM, 26-Nov-2005.)
 |- 
 -.  ph   =>    |-  ( A. x  e.  A  ph  <->  A  =  (/) )
 
Theoremraaan 3907* Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( A. x  e.  A  A. y  e.  A  (
 ph  /\  ps )  <->  (
 A. x  e.  A  ph 
 /\  A. y  e.  A  ps ) )
 
Theoremraaanv 3908* Rearrange restricted quantifiers. (Contributed by NM, 11-Mar-1997.)
 |-  ( A. x  e.  A  A. y  e.  A  ( ph  /\  ps ) 
 <->  ( A. x  e.  A  ph  /\  A. y  e.  A  ps ) )
 
Theoremsbss 3909* Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
 |-  ( [ y  /  x ] x  C_  A  <->  y 
 C_  A )
 
Theoremsbcssg 3910 Distribute proper substitution through a subclass relation. (Contributed by Alan Sare, 22-Jul-2012.) (Proof shortened by Alexander van der Vekens, 23-Jul-2017.)
 |-  ( A  e.  V  ->  ( [. A  /  x ]. B  C_  C  <->  [_ A  /  x ]_ B  C_  [_ A  /  x ]_ C ) )
 
2.1.15  "Weak deduction theorem" for set theory

In a Hilbert system of logic (which consists of a set of axioms, modus ponens, and the generalization rule), converting a deduction to a proof using the Deduction Theorem (taught in introductory logic books) involves an exponential increase of the number of steps as hypotheses are successively eliminated. Here is a trick that is not as general as the Deduction Theorem but requires only a linear increase in the number of steps.

The general problem: We want to convert a deduction P |- Q into a proof of the theorem |- P -> Q i.e. we want to eliminate the hypothesis P. Normally this is done using the Deduction (meta)Theorem, which looks at the microscopic steps of the deduction and usually doubles or triples the number of these microscopic steps for each hypothesis that is eliminated. We will look at a special case of this problem, without appealing to the Deduction Theorem.

We assume ZF with class notation. A and B are arbitrary (possibly proper) classes. P, Q, R, S and T are wffs.

We define the conditional operator, if(P, A, B), as follows: if(P, A, B) =def= { x | (x \in A & P) v (x \in B & -. P) } (where x does not occur in A, B, or P).

Lemma 1. A = if(P, A, B) -> (P <-> R), B = if(P, A, B) -> (S <-> R), S |- R Proof: Logic and Axiom of Extensionality.

Lemma 2. A = if(P, A, B) -> (Q <-> T), T |- P -> Q Proof: Logic and Axiom of Extensionality.

Here's a simple example that illustrates how it works. Suppose we have a deduction Ord A |- Tr A which means, "Assume A is an ordinal class. Then A is a transitive class." Note that A is a class variable that may be substituted with any class expression, so this is really a deduction scheme.

We want to convert this to a proof of the theorem (scheme) |- Ord A -> Tr A.

The catch is that we must be able to prove "Ord A" for at least one object A (and this is what makes it weaker than the ordinary Deduction Theorem). However, it is easy to prove |- Ord 0 (the empty set is ordinal). (For a typical textbook "theorem," i.e. deduction, there is usually at least one object satisfying each hypothesis, otherwise the theorem would not be very useful. We can always go back to the standard Deduction Theorem for those hypotheses where this is not the case.) Continuing with the example:

Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Ord A <-> Ord if(Ord A, A, 0)) (1) |- 0 = if(Ord A, A, 0) -> (Ord 0 <-> Ord if(Ord A, A, 0)) (2) From (1), (2) and |- Ord 0, Lemma 1 yields |- Ord if(Ord A, A, 0) (3) From (3) and substituting if(Ord A, A, 0) for A in the original deduction, |- Tr if(Ord A, A, 0) (4) Equality axioms (and Extensionality) yield |- A = if(Ord A, A, 0) -> (Tr A <-> Tr if(Ord A, A, 0)) (5) From (4) and (5), Lemma 2 yields |- Ord A -> Tr A (Q.E.D.)

 
Syntaxcif 3911 Extend class notation to include the conditional operator. See df-if 3912 for a description. (In older databases this was denoted "ded".)
 class  if ( ph ,  A ,  B )
 
Definitiondf-if 3912* Define the conditional operator. Read  if ( ph ,  A ,  B ) as "if  ph then  A else  B." See iftrue 3917 and iffalse 3920 for its values. In mathematical literature, this operator is rarely defined formally but is implicit in informal definitions such as "let f(x)=0 if x=0 and 1/x otherwise." (In older versions of this database, this operator was denoted "ded" and called the "deduction class.")

An important use for us is in conjunction with the weak deduction theorem, which converts a hypothesis into an antecedent. In that role,  A is a class variable in the hypothesis and  B is a class (usually a constant) that makes the hypothesis true when it is substituted for  A. See dedth 3962 for the main part of the weak deduction theorem, elimhyp 3969 to eliminate a hypothesis, and keephyp 3975 to keep a hypothesis. See the Deduction Theorem link on the Metamath Proof Explorer Home Page for a description of the weak deduction theorem. (Contributed by NM, 15-May-1999.)

 |- 
 if ( ph ,  A ,  B )  =  { x  |  ( ( x  e.  A  /\  ph )  \/  ( x  e.  B  /\  -.  ph ) ) }
 
Theoremdfif2 3913* An alternate definition of the conditional operator df-if 3912 with one fewer connectives (but probably less intuitive to understand). (Contributed by NM, 30-Jan-2006.)
 |- 
 if ( ph ,  A ,  B )  =  { x  |  ( ( x  e.  B  -> 
 ph )  ->  ( x  e.  A  /\  ph ) ) }
 
Theoremdfif6 3914* An alternate definition of the conditional operator df-if 3912 as a simple class abstraction. (Contributed by Mario Carneiro, 8-Sep-2013.)
 |- 
 if ( ph ,  A ,  B )  =  ( { x  e.  A  |  ph }  u.  { x  e.  B  |  -.  ph } )
 
Theoremifeq1 3915 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  =  B  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  C )
 )
 
Theoremifeq2 3916 Equality theorem for conditional operator. (Contributed by NM, 1-Sep-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  ( A  =  B  ->  if ( ph ,  C ,  A )  =  if ( ph ,  C ,  B )
 )
 
Theoremiftrue 3917 Value of the conditional operator when its first argument is true. (Contributed by NM, 15-May-1999.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( ph  ->  if ( ph ,  A ,  B )  =  A )
 
Theoremiftruei 3918 Inference associated with iftrue 3917. (Contributed by BJ, 7-Oct-2018.)
 |-  ph   =>    |- 
 if ( ph ,  A ,  B )  =  A
 
Theoremiftrued 3919 Value of the conditional operator when its first argument is true. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ch )   =>    |-  ( ph  ->  if ( ch ,  A ,  B )  =  A )
 
Theoremiffalse 3920 Value of the conditional operator when its first argument is false. (Contributed by NM, 14-Aug-1999.)
 |-  ( -.  ph  ->  if ( ph ,  A ,  B )  =  B )
 
Theoremiffalsei 3921 Inference associated with iffalse 3920. (Contributed by BJ, 7-Oct-2018.)
 |- 
 -.  ph   =>    |- 
 if ( ph ,  A ,  B )  =  B
 
Theoremiffalsed 3922 Value of the conditional operator when its first argument is false. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  -.  ch )   =>    |-  ( ph  ->  if ( ch ,  A ,  B )  =  B )
 
Theoremifnefalse 3923 When values are unequal, but an "if" condition checks if they are equal, then the "false" branch results. This is a simple utility to provide a slight shortening and simplification of proofs vs. applying iffalse 3920 directly in this case. It happens, e.g., in oevn0 7228. (Contributed by David A. Wheeler, 15-May-2015.)
 |-  ( A  =/=  B  ->  if ( A  =  B ,  C ,  D )  =  D )
 
Theoremifsb 3924 Distribute a function over an if-clause. (Contributed by Mario Carneiro, 14-Aug-2013.)
 |-  ( if ( ph ,  A ,  B )  =  A  ->  C  =  D )   &    |-  ( if ( ph ,  A ,  B )  =  B  ->  C  =  E )   =>    |-  C  =  if ( ph ,  D ,  E )
 
Theoremdfif3 3925* Alternate definition of the conditional operator df-if 3912. Note that  ph is independent of  x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.) (Revised by Mario Carneiro, 8-Sep-2013.)
 |-  C  =  { x  |  ph }   =>    |- 
 if ( ph ,  A ,  B )  =  ( ( A  i^i  C )  u.  ( B  i^i  ( _V  \  C ) ) )
 
Theoremdfif4 3926* Alternate definition of the conditional operator df-if 3912. Note that  ph is independent of  x i.e. a constant true or false. (Contributed by NM, 25-Aug-2013.)
 |-  C  =  { x  |  ph }   =>    |- 
 if ( ph ,  A ,  B )  =  ( ( A  u.  B )  i^i  ( ( A  u.  ( _V  \  C ) )  i^i  ( B  u.  C ) ) )
 
Theoremdfif5 3927* Alternate definition of the conditional operator df-if 3912. Note that  ph is independent of  x i.e. a constant true or false (see also abvor0 3780). (Contributed by Gérard Lang, 18-Aug-2013.)
 |-  C  =  { x  |  ph }   =>    |- 
 if ( ph ,  A ,  B )  =  ( ( A  i^i  B )  u.  ( ( ( A  \  B )  i^i  C )  u.  ( ( B  \  A )  i^i  ( _V  \  C ) ) ) )
 
Theoremifeq12 3928 Equality theorem for conditional operators. (Contributed by NM, 1-Sep-2004.)
 |-  ( ( A  =  B  /\  C  =  D )  ->  if ( ph ,  A ,  C )  =  if ( ph ,  B ,  D ) )
 
Theoremifeq1d 3929 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2d 3930 Equality deduction for conditional operator. (Contributed by NM, 16-Feb-2005.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
Theoremifeq12d 3931 Equality deduction for conditional operator. (Contributed by NM, 24-Mar-2015.)
 |-  ( ph  ->  A  =  B )   &    |-  ( ph  ->  C  =  D )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  D ) )
 
Theoremifbi 3932 Equivalence theorem for conditional operators. (Contributed by Raph Levien, 15-Jan-2004.)
 |-  ( ( ph  <->  ps )  ->  if ( ph ,  A ,  B )  =  if ( ps ,  A ,  B ) )
 
Theoremifbid 3933 Equivalence deduction for conditional operators. (Contributed by NM, 18-Apr-2005.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  A ,  B ) )
 
Theoremifbieq1d 3934 Equivalence/equality deduction for conditional operators. (Contributed by JJ, 25-Sep-2018.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ch ,  B ,  C ) )
 
Theoremifbieq2i 3935 Equivalence/equality inference for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  <->  ps )   &    |-  A  =  B   =>    |-  if ( ph ,  C ,  A )  =  if ( ps ,  C ,  B )
 
Theoremifbieq2d 3936 Equivalence/equality deduction for conditional operators. (Contributed by Paul Chapman, 22-Jun-2011.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ch ,  C ,  B ) )
 
Theoremifbieq12i 3937 Equivalence deduction for conditional operators. (Contributed by NM, 18-Mar-2013.)
 |-  ( ph  <->  ps )   &    |-  A  =  C   &    |-  B  =  D   =>    |- 
 if ( ph ,  A ,  B )  =  if ( ps ,  C ,  D )
 
Theoremifbieq12d 3938 Equivalence deduction for conditional operators. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( ph  ->  A  =  C )   &    |-  ( ph  ->  B  =  D )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  if ( ch ,  C ,  D ) )
 
Theoremnfifd 3939 Deduction version of nfif 3940. (Contributed by NM, 15-Feb-2013.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/_ x A )   &    |-  ( ph  ->  F/_ x B )   =>    |-  ( ph  ->  F/_ x if ( ps ,  A ,  B ) )
 
Theoremnfif 3940 Bound-variable hypothesis builder for a conditional operator. (Contributed by NM, 16-Feb-2005.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |- 
 F/ x ph   &    |-  F/_ x A   &    |-  F/_ x B   =>    |-  F/_ x if ( ph ,  A ,  B )
 
Theoremifeq1da 3941 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ph  /\  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  A ,  C )  =  if ( ps ,  B ,  C ) )
 
Theoremifeq2da 3942 Conditional equality. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ph  /\  -.  ps )  ->  A  =  B )   =>    |-  ( ph  ->  if ( ps ,  C ,  A )  =  if ( ps ,  C ,  B ) )
 
Theoremifclda 3943 Conditional closure. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( ph  /\  ps )  ->  A  e.  C )   &    |-  ( ( ph  /\  -.  ps )  ->  B  e.  C )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
 
Theoremifeqda 3944 Separation of the values of the conditional operator. (Contributed by Alexander van der Vekens, 13-Apr-2018.)
 |-  ( ( ph  /\  ps )  ->  A  =  C )   &    |-  ( ( ph  /\  -.  ps )  ->  B  =  C )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  =  C )
 
Theoremelimif 3945 Elimination of a conditional operator contained in a wff  ps. (Contributed by NM, 15-Feb-2005.) (Proof shortened by NM, 25-Apr-2019.)
 |-  ( if ( ph ,  A ,  B )  =  A  ->  ( ps 
 <->  ch ) )   &    |-  ( if ( ph ,  A ,  B )  =  B  ->  ( ps  <->  th ) )   =>    |-  ( ps  <->  ( ( ph  /\ 
 ch )  \/  ( -.  ph  /\  th )
 ) )
 
Theoremifbothda 3946 A wff  th containing a conditional operator is true when both of its cases are true. (Contributed by NM, 15-Feb-2015.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   &    |-  (
 ( et  /\  ph )  ->  ps )   &    |-  ( ( et 
 /\  -.  ph )  ->  ch )   =>    |-  ( et  ->  th )
 
Theoremifboth 3947 A wff  th containing a conditional operator is true when both of its cases are true. (Contributed by NM, 3-Sep-2006.) (Revised by Mario Carneiro, 15-Feb-2015.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   =>    |-  ( ( ps 
 /\  ch )  ->  th )
 
Theoremifid 3948 Identical true and false arguments in the conditional operator. (Contributed by NM, 18-Apr-2005.)
 |- 
 if ( ph ,  A ,  A )  =  A
 
Theoremeqif 3949 Expansion of an equality with a conditional operator. (Contributed by NM, 14-Feb-2005.)
 |-  ( A  =  if ( ph ,  B ,  C )  <->  ( ( ph  /\  A  =  B )  \/  ( -.  ph  /\  A  =  C ) ) )
 
Theoremifval 3950 Another expression of the value of the  if predicate, analogous to eqif 3949. See also the more specialized iftrue 3917 and iffalse 3920. (Contributed by BJ, 6-Apr-2019.)
 |-  ( A  =  if ( ph ,  B ,  C )  <->  ( ( ph  ->  A  =  B ) 
 /\  ( -.  ph  ->  A  =  C ) ) )
 
Theoremelif 3951 Membership in a conditional operator. (Contributed by NM, 14-Feb-2005.)
 |-  ( A  e.  if ( ph ,  B ,  C )  <->  ( ( ph  /\  A  e.  B )  \/  ( -.  ph  /\  A  e.  C ) ) )
 
Theoremifel 3952 Membership of a conditional operator. (Contributed by NM, 10-Sep-2005.)
 |-  ( if ( ph ,  A ,  B )  e.  C  <->  ( ( ph  /\  A  e.  C )  \/  ( -.  ph  /\  B  e.  C ) ) )
 
Theoremifcl 3953 Membership (closure) of a conditional operator. (Contributed by NM, 4-Apr-2005.)
 |-  ( ( A  e.  C  /\  B  e.  C )  ->  if ( ph ,  A ,  B )  e.  C )
 
Theoremifcld 3954 Membership (closure) of a conditional operator, deduction form. (Contributed by SO, 16-Jul-2018.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  C )   =>    |-  ( ph  ->  if ( ps ,  A ,  B )  e.  C )
 
Theoremifeqor 3955 The possible values of a conditional operator. (Contributed by NM, 17-Jun-2007.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)
 |-  ( if ( ph ,  A ,  B )  =  A  \/  if ( ph ,  A ,  B )  =  B )
 
Theoremifnot 3956 Negating the first argument swaps the last two arguments of a conditional operator. (Contributed by NM, 21-Jun-2007.)
 |- 
 if ( -.  ph ,  A ,  B )  =  if ( ph ,  B ,  A )
 
Theoremifan 3957 Rewrite a conjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |- 
 if ( ( ph  /\ 
 ps ) ,  A ,  B )  =  if ( ph ,  if ( ps ,  A ,  B ) ,  B )
 
Theoremifor 3958 Rewrite a disjunction in an if statement as two nested conditionals. (Contributed by Mario Carneiro, 28-Jul-2014.)
 |- 
 if ( ( ph  \/  ps ) ,  A ,  B )  =  if ( ph ,  A ,  if ( ps ,  A ,  B ) )
 
Theorem2if2 3959 Resolve two nested conditionals. (Contributed by Alexander van der Vekens, 27-Mar-2018.)
 |-  ( ( ph  /\  ps )  ->  D  =  A )   &    |-  ( ( ph  /\  -.  ps 
 /\  th )  ->  D  =  B )   &    |-  ( ( ph  /\ 
 -.  ps  /\  -.  th )  ->  D  =  C )   =>    |-  ( ph  ->  D  =  if ( ps ,  A ,  if ( th ,  B ,  C ) ) )
 
Theoremifcomnan 3960 Commute the conditions in two nested conditionals if both conditions are not simultaneously true. (Contributed by SO, 15-Jul-2018.)
 |-  ( -.  ( ph  /\ 
 ps )  ->  if ( ph ,  A ,  if ( ps ,  B ,  C ) )  =  if ( ps ,  B ,  if ( ph ,  A ,  C ) ) )
 
Theoremcsbif 3961 Distribute proper substitution through the conditional operator. (Contributed by NM, 24-Feb-2013.) (Revised by NM, 19-Aug-2018.)
 |-  [_ A  /  x ]_ if ( ph ,  B ,  C )  =  if ( [. A  /  x ]. ph ,  [_ A  /  x ]_ B ,  [_ A  /  x ]_ C )
 
Theoremdedth 3962 Weak deduction theorem that eliminates a hypothesis  ph, making it become an antecedent. We assume that a proof exists for  ph when the class variable  A is replaced with a specific class 
B. The hypothesis  ch should be assigned to the inference, and the inference's hypothesis eliminated with elimhyp 3969. If the inference has other hypotheses with class variable  A, these can be kept by assigning keephyp 3975 to them. For more information, see the Deduction Theorem http://us.metamath.org/mpeuni/mmdeduction.html. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  ch ) )   &    |-  ch   =>    |-  ( ph  ->  ps )
 
Theoremdedth2h 3963 Weak deduction theorem eliminating two hypotheses. This theorem is simpler to use than dedth2v 3966 but requires that each hypothesis has exactly one class variable. See also comments in dedth 3962. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ch  <->  th ) )   &    |-  ( B  =  if ( ps ,  B ,  D )  ->  ( th 
 <->  ta ) )   &    |-  ta   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremdedth3h 3964 Weak deduction theorem eliminating three hypotheses. See comments in dedth2h 3963. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( th  <->  ta ) )   &    |-  ( B  =  if ( ps ,  B ,  R )  ->  ( ta 
 <->  et ) )   &    |-  ( C  =  if ( ch ,  C ,  S )  ->  ( et  <->  ze ) )   &    |-  ze   =>    |-  (
 ( ph  /\  ps  /\  ch )  ->  th )
 
Theoremdedth4h 3965 Weak deduction theorem eliminating four hypotheses. See comments in dedth2h 3963. (Contributed by NM, 16-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ta  <->  et ) )   &    |-  ( B  =  if ( ps ,  B ,  S )  ->  ( et 
 <->  ze ) )   &    |-  ( C  =  if ( ch ,  C ,  F )  ->  ( ze  <->  si ) )   &    |-  ( D  =  if ( th ,  D ,  G )  ->  ( si  <->  rh ) )   &    |-  rh   =>    |-  (
 ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  ->  ta )
 
Theoremdedth2v 3966 Weak deduction theorem for eliminating a hypothesis with 2 class variables. Note: if the hypothesis can be separated into two hypotheses, each with one class variable, then dedth2h 3963 is simpler to use. See also comments in dedth 3962. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  th   =>    |-  ( ph  ->  ps )
 
Theoremdedth3v 3967 Weak deduction theorem for eliminating a hypothesis with 3 class variables. See comments in dedth2v 3966. (Contributed by NM, 13-Aug-1999.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ta   =>    |-  ( ph  ->  ps )
 
Theoremdedth4v 3968 Weak deduction theorem for eliminating a hypothesis with 4 class variables. See comments in dedth2v 3966. (Contributed by NM, 21-Apr-2007.) (Proof shortened by Eric Schmidt, 28-Jul-2009.)
 |-  ( A  =  if ( ph ,  A ,  R )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  S )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  T )  ->  ( th  <->  ta ) )   &    |-  ( D  =  if ( ph ,  D ,  U )  ->  ( ta  <->  et ) )   &    |-  et   =>    |-  ( ph  ->  ps )
 
Theoremelimhyp 3969 Eliminate a hypothesis containing class variable  A when it is known for a specific class  B. For more information, see comments in dedth 3962. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ph  <->  ps ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <->  ps ) )   &    |-  ch   =>    |-  ps
 
Theoremelimhyp2v 3970 Eliminate a hypothesis containing 2 class variables. (Contributed by NM, 14-Aug-1999.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ph  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  A ,  C )  ->  ( ta  <->  et ) )   &    |-  ( D  =  if ( ph ,  B ,  D )  ->  ( et  <->  th ) )   &    |-  ta   =>    |-  th
 
Theoremelimhyp3v 3971 Eliminate a hypothesis containing 3 class variables. (Contributed by NM, 14-Aug-1999.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze ) )   &    |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si ) )   &    |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta ) )   &    |-  et   =>    |-  ta
 
Theoremelimhyp4v 3972 Eliminate a hypothesis containing 4 class variables (for use with the weak deduction theorem dedth 3962). (Contributed by NM, 16-Apr-2005.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( ph  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ( F  =  if ( ph ,  F ,  G )  ->  ( ta  <->  ps ) )   &    |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze ) )   &    |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si ) )   &    |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  rh ) )   &    |-  ( G  =  if ( ph ,  F ,  G )  ->  ( rh  <->  ps ) )   &    |-  et   =>    |-  ps
 
Theoremelimel 3973 Eliminate a membership hypothesis for weak deduction theorem, when special case  B  e.  C is provable. (Contributed by NM, 15-May-1999.)
 |-  B  e.  C   =>    |-  if ( A  e.  C ,  A ,  B )  e.  C
 
Theoremelimdhyp 3974 Version of elimhyp 3969 where the hypothesis is deduced from the final antecedent. See ghomgrplem 30315 for an example of its use. (Contributed by Paul Chapman, 25-Mar-2008.)
 |-  ( ph  ->  ps )   &    |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( th  <->  ch ) )   &    |-  th   =>    |-  ch
 
Theoremkeephyp 3975 Transform a hypothesis  ps that we want to keep (but contains the same class variable  A used in the eliminated hypothesis) for use with the weak deduction theorem. (Contributed by NM, 15-May-1999.)
 |-  ( A  =  if ( ph ,  A ,  B )  ->  ( ps  <->  th ) )   &    |-  ( B  =  if ( ph ,  A ,  B )  ->  ( ch 
 <-> 
 th ) )   &    |-  ps   &    |-  ch   =>    |-  th
 
Theoremkeephyp2v 3976 Keep a hypothesis containing 2 class variables (for use with the weak deduction theorem dedth 3962). (Contributed by NM, 16-Apr-2005.)
 |-  ( A  =  if ( ph ,  A ,  C )  ->  ( ps  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  D )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  A ,  C )  ->  ( ta  <->  et ) )   &    |-  ( D  =  if ( ph ,  B ,  D )  ->  ( et  <->  th ) )   &    |-  ps   &    |-  ta   =>    |-  th
 
Theoremkeephyp3v 3977 Keep a hypothesis containing 3 class variables. (Contributed by NM, 27-Sep-1999.)
 |-  ( A  =  if ( ph ,  A ,  D )  ->  ( rh  <->  ch ) )   &    |-  ( B  =  if ( ph ,  B ,  R )  ->  ( ch 
 <-> 
 th ) )   &    |-  ( C  =  if ( ph ,  C ,  S )  ->  ( th  <->  ta ) )   &    |-  ( D  =  if ( ph ,  A ,  D )  ->  ( et  <->  ze ) )   &    |-  ( R  =  if ( ph ,  B ,  R )  ->  ( ze  <->  si ) )   &    |-  ( S  =  if ( ph ,  C ,  S )  ->  ( si  <->  ta ) )   &    |-  rh   &    |-  et   =>    |-  ta
 
Theoremkeepel 3978 Keep a membership hypothesis for weak deduction theorem, when special case  B  e.  C is provable. (Contributed by NM, 14-Aug-1999.)
 |-  A  e.  C   &    |-  B  e.  C   =>    |- 
 if ( ph ,  A ,  B )  e.  C
 
Theoremifex 3979 Conditional operator existence. (Contributed by NM, 2-Sep-2004.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |- 
 if ( ph ,  A ,  B )  e.  _V
 
Theoremifexg 3980 Conditional operator existence. (Contributed by NM, 21-Mar-2011.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  if ( ph ,  A ,  B )  e.  _V )
 
2.1.16  Power classes
 
Syntaxcpw 3981 Extend class notation to include power class. (The tilde in the Metamath token is meant to suggest the calligraphic font of the P.)
 class  ~P A
 
Theorempwjust 3982* Soundness justification theorem for df-pw 3983. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 { x  |  x  C_  A }  =  {
 y  |  y  C_  A }
 
Definitiondf-pw 3983* Define power class. Definition 5.10 of [TakeutiZaring] p. 17, but we also let it apply to proper classes, i.e. those that are not members of  _V. When applied to a set, this produces its power set. A power set of S is the set of all subsets of S, including the empty set and S itself. For example, if  A  =  { 3 ,  5 ,  7 }, then  ~P A  =  { (/) ,  { 3 } ,  { 5 } ,  { 7 } ,  { 3 ,  5 } ,  { 3 ,  7 } ,  {
5 ,  7 } ,  { 3 ,  5 ,  7 } } (ex-pw 25877). We will later introduce the Axiom of Power Sets ax-pow 4602, which can be expressed in class notation per pwexg 4608. Still later we will prove, in hashpw 12612, that the size of the power set of a finite set is 2 raised to the power of the size of the set. (Contributed by NM, 24-Jun-1993.)
 |- 
 ~P A  =  { x  |  x  C_  A }
 
Theorempweq 3984 Equality theorem for power class. (Contributed by NM, 21-Jun-1993.)
 |-  ( A  =  B  ->  ~P A  =  ~P B )
 
Theorempweqi 3985 Equality inference for power class. (Contributed by NM, 27-Nov-2013.)
 |-  A  =  B   =>    |-  ~P A  =  ~P B
 
Theorempweqd 3986 Equality deduction for power class. (Contributed by NM, 27-Nov-2013.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ~P A  =  ~P B )
 
Theoremelpw 3987 Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
 |-  A  e.  _V   =>    |-  ( A  e.  ~P B  <->  A  C_  B )
 
Theoremselpw 3988* Setvar variable membership in a power class (common case). See elpw 3987. (Contributed by David A. Wheeler, 8-Dec-2018.)
 |-  ( x  e.  ~P A 
 <->  x  C_  A )
 
Theoremelpwg 3989 Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g 4587. (Contributed by NM, 6-Aug-2000.)
 |-  ( A  e.  V  ->  ( A  e.  ~P B 
 <->  A  C_  B )
 )
 
Theoremelpwi 3990 Subset relation implied by membership in a power class. (Contributed by NM, 17-Feb-2007.)
 |-  ( A  e.  ~P B  ->  A  C_  B )
 
Theoremelpwid 3991 An element of a power class is a subclass. Deduction form of elpwi 3990. (Contributed by David Moews, 1-May-2017.)
 |-  ( ph  ->  A  e.  ~P B )   =>    |-  ( ph  ->  A 
 C_  B )
 
Theoremelelpwi 3992 If  A belongs to a part of  C then  A belongs to  C. (Contributed by FL, 3-Aug-2009.)
 |-  ( ( A  e.  B  /\  B  e.  ~P C )  ->  A  e.  C )
 
Theoremnfpw 3993 Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.)
 |-  F/_ x A   =>    |-  F/_ x ~P A
 
Theorempwidg 3994 Membership of the original in a power set. (Contributed by Stefan O'Rear, 1-Feb-2015.)
 |-  ( A  e.  V  ->  A  e.  ~P A )
 
Theorempwid 3995 A set is a member of its power class. Theorem 87 of [Suppes] p. 47. (Contributed by NM, 5-Aug-1993.)
 |-  A  e.  _V   =>    |-  A  e.  ~P A
 
Theorempwss 3996* Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
 |-  ( ~P A  C_  B 
 <-> 
 A. x ( x 
 C_  A  ->  x  e.  B ) )
 
2.1.17  Unordered and ordered pairs
 
Theoremsnjust 3997* Soundness justification theorem for df-sn 3999. (Contributed by Rodolfo Medina, 28-Apr-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
 |- 
 { x  |  x  =  A }  =  {
 y  |  y  =  A }
 
Syntaxcsn 3998 Extend class notation to include singleton.
 class  { A }
 
Definitiondf-sn 3999* Define the singleton of a class. Definition 7.1 of [Quine] p. 48. For convenience, it is well-defined for proper classes, i.e., those that are not elements of  _V, although it is not very meaningful in this case. For an alternate definition see dfsn2 4011. (Contributed by NM, 21-Jun-1993.)
 |- 
 { A }  =  { x  |  x  =  A }
 
Syntaxcpr 4000 Extend class notation to include unordered pair.
 class  { A ,  B }
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