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Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorema16g-o 2001* A generalization of axiom ax-16 1926. Version of a16g 2000 using ax-10o 1835. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph )
 )
 
Theorema16gb 2002* A generalization of axiom ax-16 1926. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <-> 
 A. z ph )
 )
 
Theorema16nf 2003* If dtru 4095 is false, then there is only one element in the universe, so everything satisfies  F/. (Contributed by Mario Carneiro, 7-Oct-2016.)
 |-  ( A. x  x  =  y  ->  F/ z ph )
 
Theoremalbidv 2004* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. x ch )
 )
 
Theoremexbidv 2005* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. x ch )
 )
 
Theorem2albidv 2006* Formula-building rule for 2 universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  (
 A. x A. y ps 
 <-> 
 A. x A. y ch ) )
 
Theorem2exbidv 2007* Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps 
 <-> 
 E. x E. y ch ) )
 
Theorem3exbidv 2008* Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y E. z ps  <->  E. x E. y E. z ch ) )
 
Theorem4exbidv 2009* Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y E. z E. w ps  <->  E. x E. y E. z E. w ch ) )
 
Theorem19.9v 2010* Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.)
 |-  ( E. x ph  <->  ph )
 
Theorem19.21v 2011* Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as  F/ x ph in 19.21 1771 via the use of distinct variable conditions combined with nfv 1629. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 2120 derived from df-eu 2118. The "f" stands for "not free in" which is less restrictive than "does not occur in." (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x (
 ph  ->  ps )  <->  ( ph  ->  A. x ps ) )
 
Theoremalrimiv 2012* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x ps )
 
Theoremalrimivv 2013* Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( ph  ->  A. x A. y ps )
 
Theoremalrimdv 2014* Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( ps  ->  A. x ch ) )
 
Theoremnfdv 2015* Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |-  ( ph  ->  ( ps  ->  A. x ps )
 )   =>    |-  ( ph  ->  F/ x ps )
 
Theorem2ax17 2016* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
 |-  ( ph  ->  A. x A. y ph )
 
Theoremalimdv 2017* Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x ps  ->  A. x ch ) )
 
Theoremeximdv 2018* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  E. x ch ) )
 
Theorem2alimdv 2019* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  (
 A. x A. y ps  ->  A. x A. y ch ) )
 
Theorem2eximdv 2020* Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps  ->  E. x E. y ch ) )
 
Theorem19.23v 2021* Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)
 |-  ( A. x (
 ph  ->  ps )  <->  ( E. x ph 
 ->  ps ) )
 
Theorem19.23vv 2022* Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
 |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x E. y ph  ->  ps )
 )
 
Theoremexlimiv 2023* Inference from Theorem 19.23 of [Margaris] p. 90.

This inference, along with our many variants such as rexlimdv 2628, is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.mathsci.appstate.edu/~jlh/primer/hirst.pdf.

In informal proofs, the statement "Let  C be an element such that..." almost always means an implicit application of Rule C.

In essence, Rule C states that if we can prove that some element  x exists satisfying a wff, i.e.  E. x ph ( x ) where  ph ( x ) has  x free, then we can use  ph ( C ) as a hypothesis for the proof where  C is a new (ficticious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier.

We cannot do this in Metamath directly. Instead, we use the original  ph (containing  x) as an antecedent for the main part of the proof. We eventually arrive at  ( ph  ->  ps ) where  ps is the theorem to be proved and does not contain  x. Then we apply exlimiv 2023 to arrive at  ( E. x ph  ->  ps ). Finally, we separately prove  E. x ph and detach it with modus ponens ax-mp 10 to arrive at the final theorem  ps. (Contributed by NM, 5-Aug-1993.)

 |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theorempm11.53 2024* Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
 |-  ( A. x A. y ( ph  ->  ps )  <->  ( E. x ph 
 ->  A. y ps )
 )
 
Theoremexlimivv 2025* Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
 |-  ( ph  ->  ps )   =>    |-  ( E. x E. y ph  ->  ps )
 
Theoremexlimdvv 2026* Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x E. y ps  ->  ch ) )
 
Theorem19.27v 2027* Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  ps ) )
 
Theorem19.28v 2028* Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 ph  /\  A. x ps ) )
 
Theorem19.36v 2029* Special case of Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
 |-  ( E. x (
 ph  ->  ps )  <->  ( A. x ph 
 ->  ps ) )
 
Theorem19.36aiv 2030* Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theorem19.12vv 2031* Special case of 19.12 1766 where its converse holds. (Contributed by NM, 18-Jul-2001.) (Revised by Andrew Salmon, 11-Jul-2011.)
 |-  ( E. x A. y ( ph  ->  ps )  <->  A. y E. x ( ph  ->  ps )
 )
 
Theorem19.37v 2032* Special case of Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x (
 ph  ->  ps )  <->  ( ph  ->  E. x ps ) )
 
Theorem19.37aiv 2033* Inference from Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x ( ph  ->  ps )   =>    |-  ( ph  ->  E. x ps )
 
Theorem19.41v 2034* Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
 
Theorem19.41vv 2035* Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <->  ( E. x E. y ph  /\  ps )
 )
 
Theorem19.41vvv 2036* Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps )  <->  ( E. x E. y E. z ph  /\ 
 ps ) )
 
Theorem19.41vvvv 2037* Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
 |-  ( E. w E. x E. y E. z
 ( ph  /\  ps )  <->  ( E. w E. x E. y E. z ph  /\ 
 ps ) )
 
Theorem19.42v 2038* Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x (
 ph  /\  ps )  <->  (
 ph  /\  E. x ps ) )
 
Theoremexdistr 2039* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <-> 
 E. x ( ph  /\ 
 E. y ps )
 )
 
Theorem19.42vv 2040* Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <->  ( ph  /\  E. x E. y ps )
 )
 
Theorem19.42vvv 2041* Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps )  <->  ( ph  /\  E. x E. y E. z ps ) )
 
Theoremexdistr2 2042* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps )  <->  E. x ( ph  /\ 
 E. y E. z ps ) )
 
Theorem3exdistr 2043* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps  /\  ch )  <->  E. x ( ph  /\  E. y ( ps  /\  E. z ch ) ) )
 
Theorem4exdistr 2044* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z E. w ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  E. x ( ph  /\ 
 E. y ( ps 
 /\  E. z ( ch 
 /\  E. w th )
 ) ) )
 
Theoremcbvalv 2045* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph  <->  A. y ps )
 
Theoremcbvexv 2046* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E. x ph  <->  E. y ps )
 
Theoremcbval2 2047* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x A. y ph  <->  A. z A. w ps )
 
Theoremcbvex2 2048* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   &    |-  F/ x ps   &    |-  F/ y ps   &    |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x E. y ph  <->  E. z E. w ps )
 
Theoremcbval2v 2049* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
 |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x A. y ph  <->  A. z A. w ps )
 
Theoremcbvex2v 2050* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. x E. y ph  <->  E. z E. w ps )
 
Theoremcbvald 2051* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2092. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  (
 A. x ps  <->  A. y ch )
 )
 
Theoremcbvexd 2052* Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2092. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ph  ->  F/ y ps )   &    |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( E. x ps  <->  E. y ch )
 )
 
Theoremcbvex4v 2053* Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( ( x  =  v  /\  y  =  u )  ->  ( ph 
 <->  ps ) )   &    |-  (
 ( z  =  f 
 /\  w  =  g )  ->  ( ps  <->  ch ) )   =>    |-  ( E. x E. y E. z E. w ph  <->  E. v E. u E. f E. g ch )
 
Theoremeean 2054 Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x E. y (
 ph  /\  ps )  <->  ( E. x ph  /\  E. y ps ) )
 
Theoremeeanv 2055* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <->  ( E. x ph  /\ 
 E. y ps )
 )
 
Theoremeeeanv 2056* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps  /\  ch )  <->  ( E. x ph  /\  E. y ps  /\  E. z ch ) )
 
Theoremee4anv 2057* Rearrange existential quantifiers. (Contributed by NM, 31-Jul-1995.)
 |-  ( E. x E. y E. z E. w ( ph  /\  ps )  <->  ( E. x E. y ph  /\  E. z E. w ps ) )
 
Theoremnexdv 2058* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremchvarv 2059* Implicit substitution of  y for  x into a theorem. (Contributed by NM, 20-Apr-1994.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ph   =>    |- 
 ps
 
Theoremcleljust 2060* When the class variables in definition df-clel 2249 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1622 with the class variables in wcel 1621. Note: This proof is referenced on the Metamath Proof Explorer Home Page and shouldn't be changed. (Contributed by NM, 28-Jan-2004.) (Revised by NM, 10-Jan-2017.) (Proof modification is discouraged.)
 |-  ( x  e.  y  <->  E. z ( z  =  x  /\  z  e.  y ) )
 
1.6.5  More substitution theorems
 
Theoremequsb3lem 2061* Lemma for equsb3 2062. (Contributed by Raph Levien and FL, 4-Dec-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  =  z  <-> 
 x  =  z )
 
Theoremequsb3 2062* Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
 |-  ( [ x  /  y ] y  =  z  <-> 
 x  =  z )
 
Theoremelsb3 2063* Substitution applied to an atomic membership wff. (Contributed by NM, 7-Nov-2006.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] y  e.  z  <->  x  e.  z )
 
Theoremelsb4 2064* Substitution applied to an atomic membership wff. (Contributed by Rodolfo Medina, 3-Apr-2010.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( [ x  /  y ] z  e.  y  <->  z  e.  x )
 
Theoremhbs1 2065*  x is not free in  [ y  /  x ] ph when  x and  y are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 
Theoremnfs1v 2066*  x is not free in  [ y  /  x ] ph when  x and  y are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ x [ y  /  x ] ph
 
Theoremsbhb 2067* Two ways of expressing " x is (effectively) not free in  ph." (Contributed by NM, 29-May-2009.)
 |-  ( ( ph  ->  A. x ph )  <->  A. y ( ph  ->  [ y  /  x ] ph ) )
 
Theoremsbnf2 2068* Two ways of expressing " x is (effectively) not free in  ph." (Contributed by Gérard Lang, 14-Nov-2013.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |-  ( F/ x ph  <->  A. y A. z ( [
 y  /  x ] ph 
 <->  [ z  /  x ] ph ) )
 
Theoremnfsb 2069* If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  y and  z are distinct. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z ph   =>    |- 
 F/ z [ y  /  x ] ph
 
Theoremhbsb 2070* If  z is not free in  ph, it is not free in  [ y  /  x ] ph when  y and  z are distinct. (Contributed by NM, 12-Aug-1993.)
 |-  ( ph  ->  A. z ph )   =>    |-  ( [ y  /  x ] ph  ->  A. z [ y  /  x ] ph )
 
Theoremnfsbd 2071* Deduction version of nfsb 2069. (Contributed by NM, 15-Feb-2013.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ z ps )   =>    |-  ( ph  ->  F/ z [ y  /  x ] ps )
 
Theorem2sb5 2072* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  E. x E. y
 ( ( x  =  z  /\  y  =  w )  /\  ph )
 )
 
Theorem2sb6 2073* Equivalence for double substitution. (Contributed by NM, 3-Feb-2005.)
 |-  ( [ z  /  x ] [ w  /  y ] ph  <->  A. x A. y
 ( ( x  =  z  /\  y  =  w )  ->  ph )
 )
 
Theoremsbcom2 2074* Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 27-May-1997.)
 |-  ( [ w  /  z ] [ y  /  x ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theorempm11.07 2075* Theorem *11.07 in [WhiteheadRussell] p. 159. (Contributed by Andrew Salmon, 17-Jun-2011.)
 |-  ( [ w  /  x ] [ y  /  z ] ph  <->  [ y  /  x ] [ w  /  z ] ph )
 
Theoremsb6a 2076* Equivalence for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  [ x  /  y ] ph )
 )
 
Theorem2sb5rf 2077* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   =>    |-  ( ph 
 <-> 
 E. z E. w ( ( z  =  x  /\  w  =  y )  /\  [
 z  /  x ] [ w  /  y ] ph ) )
 
Theorem2sb6rf 2078* Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   &    |-  F/ w ph   =>    |-  ( ph 
 <-> 
 A. z A. w ( ( z  =  x  /\  w  =  y )  ->  [ z  /  x ] [ w  /  y ] ph )
 )
 
Theoremdfsb7 2079* An alternate definition of proper substitution df-sb 1883. By introducing a dummy variable  z in the definiens, we are able to eliminate any distinct variable restrictions among the variables  x,  y, and  ph of the definiendum. No distinct variable conflicts arise because  z effectively insulates  x from  y. To achieve this, we use a chain of two substitutions in the form of sb5 1993, first  z for  x then  y for  z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 2240. Theorem sb7h 2081 provides a version where  ph and  z don't have to be distinct. (Contributed by NM, 28-Jan-2004.)
 |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb7f 2080* This version of dfsb7 2079 does not require that  ph and  z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1628 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1883 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ z ph   =>    |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb7h 2081* This version of dfsb7 2079 does not require that  ph and  z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1628 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1883 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.) (Contributed by NM, 26-Jul-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( ph  ->  A. z ph )   =>    |-  ( [ y  /  x ] ph  <->  E. z ( z  =  y  /\  E. x ( x  =  z  /\  ph )
 ) )
 
Theoremsb10f 2082* Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived. (Contributed by NM, 9-May-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)
 |- 
 F/ x ph   =>    |-  ( [ y  /  z ] ph  <->  E. x ( x  =  y  /\  [ x  /  z ] ph ) )
 
Theoremsbid2v 2083* An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
 
Theoremsbelx 2084* Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  E. x ( x  =  y  /\  [ x  /  y ] ph ) )
 
Theoremsbel2x 2085* Elimination of double substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  <->  E. x E. y
 ( ( x  =  z  /\  y  =  w )  /\  [
 y  /  w ] [ x  /  z ] ph ) )
 
Theoremsbal1 2086* A theorem used in elimination of disjoint variable restriction on  x and  y by replacing it with a distinctor  -.  A. x x  =  z. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  z  ->  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 )
 
Theoremsbal 2087* Move universal quantifier in and out of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 
Theoremsbex 2088* Move existential quantifier in and out of substitution. (Contributed by NM, 27-Sep-2003.)
 |-  ( [ z  /  y ] E. x ph  <->  E. x [ z  /  y ] ph )
 
Theoremsbalv 2089* Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.)
 |-  ( [ y  /  x ] ph  <->  ps )   =>    |-  ( [ y  /  x ] A. z ph  <->  A. z ps )
 
Theoremexsb 2090* An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
 |-  ( E. x ph  <->  E. y A. x ( x  =  y  ->  ph )
 )
 
Theorem2exsb 2091* An equivalent expression for double existence. (Contributed by NM, 2-Feb-2005.)
 |-  ( E. x E. y ph  <->  E. z E. w A. x A. y ( ( x  =  z 
 /\  y  =  w )  ->  ph ) )
 
Theoremdvelim 2092* This theorem can be used to eliminate a distinct variable restriction on  x and  z and replace it with the "distinctor"  -.  A. x x  =  y as an antecedent.  ph normally has  z free and can be read  ph ( z ), and  ps substitutes  y for  z and can be read  ph ( y ). We don't require that 
x and  y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with  A. x A. z, conjoin them, and apply dvelimdf 1976.

Other variants of this theorem are dvelimh 1974 (with no distinct variable restrictions), dvelimfALT 1853 (that avoids ax-11 1624), and dvelimALT 2094 (that avoids ax-10 1678). (Contributed by NM, 23-Nov-1994.)

 |-  ( ph  ->  A. x ph )   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdvelimv 2093* This theorem can be used to eliminate a distinct variable restriction on  x and  z and replace it with the "distinctor"  -.  A. x x  =  y as an antecedent.  ph normally has  z free and can be read  ph ( z ), and  ps substitutes  y for  z and can be read  ph ( y ). We don't require that 
x and  y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent.

To obtain a closed-theorem form of this inference, prefix the hypotheses with  A. x A. z, conjoin them, and apply dvelimdf 1976.

Other variants of this theorem are dvelimf 1975 (with no distinct variable restrictions), dvelimfALT 1853 (that avoids ax-11 1624), and dvelimALT 2094 (that avoids ax-10 1678). (Contributed by NM, 23-Nov-1994.)

 |- 
 F/ x ph   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x ps )
 
TheoremdvelimALT 2094* Version of dvelim 2092 that doesn't use ax-10 1678. (See dvelimfALT 1853 for a version that doesn't use ax-11 1624.) (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdveeq1 2095* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremdveeq1-o 2096* Quantifier introduction when one pair of variables is distinct. Version of dveeq1 2095 using ax-10o . (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremdveeq1ALT 2097* Version of dveeq1 2095 using ax-16 1926 instead of ax-17 1628. (Contributed by NM, 29-Apr-2008.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremdveel1 2098* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( y  e.  z  ->  A. x  y  e.  z ) )
 
Theoremdveel2 2099* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( z  e.  y  ->  A. x  z  e.  y ) )
 
Theoremsbal2 2100* Move quantifier in and out of substitution. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ( [ z  /  y ] A. x ph  <->  A. x [ z  /  y ] ph )
 )
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