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Theorem List for Metamath Proof Explorer - 2001-2100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremequsalh 2001 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( A. x ( x  =  y  ->  ph )  <->  ps )
 
Theoremequsex 2002 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Feb-2018.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
TheoremequsexOLD 2003 Obsolete proof of equsex 2002 as of 6-Feb-2018. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
Theoremequsexh 2004 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( ps  ->  A. x ps )   &    |-  ( x  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  ps )
 
Theoremax12olem1 2005* Lemma for nfeqf 2009 and dveeq1 2021. Used to eliminate distinct variable constraints. The proof of ax12o 2010 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Feb-2018.)
 |-  ( y  =  z  <->  A. w ( y  =  w  ->  z  =  w ) )
 
Theoremax12olem2 2006* Lemma for nfeqf 2009 and dveeq1 2021. This lemma is equivalent to ax12v 1951 with one distinct variable constraint removed. (Contributed by Wolf Lammen, 29-Apr-2018.)
 |-  ( -.  x  =  y  ->  ( E. x  y  =  z  ->  y  =  z ) )
 
Theoremax12olem3 2007* Lemma for nfeqf 2009 and dveeq1 2021. Convert ax12olem2 2006 into a more general form. (Contributed by Wolf Lammen, 29-Apr-2018.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x  y  =  z )
 
Theoremax12olem4 2008* Lemma for nfeqf 2009. A technical step to remove a distinct variable constraint from ax12v 1951. (Contributed by Wolf Lammen, 29-Apr-2018.)
 |-  ( ph  ->  F/ x  y  =  w )   &    |-  ( ps  ->  F/ x  z  =  w )   =>    |-  ( ( ph  /\ 
 ps )  ->  F/ x  y  =  z )
 
Theoremnfeqf 2009 A variable is effectively not free in an equality if it is not either of the involved variables.  F/ version of ax-12o 2219. (Contributed by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)
 |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
 
Theoremax12o 2010 Derive set.mm's original ax-12o 2219 from the shorter ax-12 1950. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 29-Apr-2018.)
 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theoremax12olem1OLD 2011* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Similar to equvin 2087 but with a negated equality. (Contributed by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 20-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E. w ( y  =  w  /\  -.  z  =  w )  <->  -.  y  =  z
 )
 
Theoremax12olem2OLD 2012* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Negate the equalities in ax-12 1950, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.) (Proof shortened by Wolf Lammen, 23-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w )
 )   =>    |-  ( -.  x  =  y  ->  ( -.  y  =  z  ->  A. x  -.  y  =  z ) )
 
Theoremax12olem3OLD 2013 Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Show the equivalence of an intermediate equivalent to ax12o 2010 with the conjunction of ax-12 1950 and a variant with negated equalities. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( -.  x  =  y  ->  ( -. 
 A. x  -.  y  =  z  ->  A. x  y  =  z )
 ) 
 <->  ( ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )  /\  ( -.  x  =  y  ->  ( -.  y  =  z 
 ->  A. x  -.  y  =  z ) ) ) )
 
Theoremax12olem4OLD 2014* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Construct an intermediate equivalent to ax-12 1950 from two instances of ax-12 1950. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )   &    |-  ( -.  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w )
 )   =>    |-  ( -.  x  =  y  ->  ( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )
 
Theoremax12olem5OLD 2015 Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. See ax12olem6OLD 2016 for derivation of ax12oOLD 2018 from the conclusion. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  x  =  y  ->  ( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )   =>    |-  ( -.  A. x  x  =  y  ->  (
 y  =  z  ->  A. x  y  =  z ) )
 
Theoremax12olem6OLD 2016* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Derivation of ax12oOLD 2018 from the hypotheses, without using ax12oOLD 2018. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  z  ->  ( z  =  w  ->  A. x  z  =  w ) )   &    |-  ( -.  A. x  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w )
 )   =>    |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  (
 y  =  z  ->  A. x  y  =  z ) ) )
 
Theoremax12olem7OLD 2017* Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. Lemma for ax12oOLD 2018. Derivation of ax12oOLD 2018 from the hypotheses, without using ax12oOLD 2018. (Contributed by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  x  =  z  ->  ( -.  A. x  -.  z  =  w  ->  A. x  z  =  w ) )   &    |-  ( -.  x  =  y 
 ->  ( -.  A. x  -.  y  =  w  ->  A. x  y  =  w ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( -.  A. x  x  =  z  ->  ( y  =  z  ->  A. x  y  =  z ) ) )
 
Theoremax12oOLD 2018 Obsolete proof of ax12oOLD 2018 as of 30-Jan-2018. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theoremax12 2019 Derive ax-12 1950 from ax12v 1951 via ax12o 2010. This shows that the weakening in ax12v 1951 is still sufficient for a complete system. (Contributed by NM, 21-Dec-2015.) (Proof shortened by Wolf Lammen, 31-Jan-2018.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax12OLD 2020 Obsolete proof of ax12 2019 as of 31-Jan-2018. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremdveeq1 2021* Quantifier introduction when one pair of variables is distinct. Revised to be independent of dvelimv 2074. (Contributed by NM, 2-Jan-2002.) (Revised by Wolf Lammen, 29-Apr-2018.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremax10lem1 2022* Lemma for ax10 2025. Change bound variable. (Contributed by NM, 22-Jul-2015.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  w )
 
Theoremax10lem2 2023* Lemma for ax10 2025. Change bound variable. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  x )
 
Theoremax10o2 2024 Same as ax10o 2038 but with reversed antecedent. (Contributed by NM, 25-Jul-2015.)
 |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph ) )
 
Theoremax10 2025 Derive set.mm's original ax-10 2217 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremax10lem2OLD 2026* Obsolete proof of a lemma for ax10 2025 as of 17-Feb-2018. Change free variable. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  A. x  x  =  z )
 
Theoremax10lem3OLD 2027* Obsolete proof of a lemma for ax10 2025 as of 17-Feb-2018. Similar to ax-10 2217 but with distinct variables. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
TheoremdvelimvOLD 2028* Obsolete proof of dvelimv 2074 as of 17-Feb-2018. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( z  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdveeq2OLD 2029* Obsolete proof of dveeq2 2077 as of 25-Feb-2018. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y ) )
 
Theoremax10lem4OLD 2030* Obsolete proof of ax10lem2 2023 as of 17-Feb-2018. (Contributed by NM, 8-Jul-2016.) (New usage is discouraged.) (Proof modification is discouraged. )
 |-  ( A. x  x  =  w  ->  A. y  y  =  x )
 
Theoremax10lem5OLD 2031* Obsolete proof of ax10o2 2024 as of 17-Feb-2018. (Contributed by NM, 22-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. z  z  =  w  ->  A. y  y  =  x )
 
Theoremax10OLD 2032 Obsolete proof of ax10 2025 as of 17-Feb-2018. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremax9OLD 2033 Obsolete proof of ax9 1953 as of 4-Feb-2018. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modfication is discouraged.)
 |- 
 -.  A. x  -.  x  =  y
 
Theorema9eOLD 2034 Obsolete proof of a9e 1952 as of 4-Feb-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modfication is discouraged.)
 |- 
 E. x  x  =  y
 
Theoremaecom 2035 Commutation law for identical variable specifiers. The antecedent and consequent are true when  x and  y are substituted with the same variable. Lemma L12 in [Megill] p. 445 (p. 12 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremaecoms 2036 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theoremnaecoms 2037 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  ->  ph )
 
Theoremax10o 2038 Show that ax-10o 2216 can be derived from ax-10 2217 in the form of ax10 2025. Normally, ax10o 2038 should be used rather than ax-10o 2216, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
 |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph ) )
 
Theoremax10oOLD 2039 Obsolete proof of ax10o 2038 as of 21-Apr-2018. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph ) )
 
Theoremhbae 2040 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
TheoremhbaeOLD 2041 Obsolete proof of hbae 2040 as of 21-Apr-2018. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremnfae 2042 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z A. x  x  =  y
 
Theoremhbnae 2043 All variables are effectively bound in a distinct variable specifier. Lemma L19 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  A. z  -.  A. x  x  =  y )
 
Theoremnfnae 2044 All variables are effectively bound in a distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z  -.  A. x  x  =  y
 
Theoremhbnaes 2045 Rule that applies hbnae 2043 to antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. z  -.  A. x  x  =  y 
 ->  ph )   =>    |-  ( -.  A. x  x  =  y  ->  ph )
 
Theoremaevlem1 2046* Lemma for aev 2047 and a16g 2048. Change free and bound variables. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.)
 |-  ( A. z  z  =  w  ->  A. y  y  =  x )
 
Theoremaev 2047* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.)
 |-  ( A. x  x  =  y  ->  A. z  w  =  v )
 
Theorema16g 2048* Generalization of ax16 2050. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 18-Feb-2018.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph )
 )
 
Theorema16gOLD 2049* Obsolete proof of a16g 2048 as of 18-Feb-2018. (Contributed by NM, 25-Jul-2015.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph )
 )
 
Theoremax16 2050* Proof of older axiom ax-16 2221. (Contributed by NM, 8-Nov-2006.) (Revised by NM, 22-Sep-2017.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph )
 )
 
Theoremax16i 2051* Inference with ax16 2050 as its conclusion. (Contributed by NM, 20-May-2008.) (Proof modification is discouraged.)
 |-  ( x  =  z 
 ->  ( ph  <->  ps ) )   &    |-  ( ps  ->  A. x ps )   =>    |-  ( A. x  x  =  y  ->  ( ph  ->  A. x ph ) )
 
Theorema16gb 2052* A generalization of axiom ax-16 2221. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <-> 
 A. z ph )
 )
 
Theorema16nf 2053* If dtru 4390 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 )
 
TheoremnfeqfOLD 2054 Obsolete proof of nfeqf 2009 as of 29-Apr-2018. (Contributed by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( -.  A. z  z  =  x  /\  -.  A. z  z  =  y )  ->  F/ z  x  =  y )
 
Theoremdral2 2055 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.) (Revised by Wolf Lammen, 4-Mar-2018.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdral2OLD 2056 Obsolete proof of dral2 2055 as of 4-Mar-2018. (Contributed by NM, 27-Feb-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. z ph  <->  A. z ps )
 )
 
Theoremdral1 2057 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 24-Nov-1994.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. y ps )
 )
 
Theoremdral1OLD 2058 Obsolete proof of dral1 2057 as of 4-Mar-2018. (Contributed by NM, 24-Nov-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  (
 A. x ph  <->  A. y ps )
 )
 
Theoremdrex1 2059 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. x ph  <->  E. y ps )
 )
 
Theoremdrex2 2060 Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( E. z ph  <->  E. z ps )
 )
 
Theoremdrnf1 2061 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ x ph  <->  F/ y ps )
 )
 
Theoremdrnf2 2062 Formula-building lemma for use with the Distinctor Reduction Theorem. (Contributed by Mario Carneiro, 4-Oct-2016.) (Proof shortened by Wolf Lammen, 5-May-2018.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theoremdrnf2OLD 2063 Obsolete proof of drnf2 2062 as of 5-May-2018. (Contributed by Mario Carneiro, 4-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( ph 
 <->  ps ) )   =>    |-  ( A. x  x  =  y  ->  ( F/ z ph  <->  F/ z ps )
 )
 
Theoremnfald2 2064 Variation on nfald 1871 which adds the hypothesis that  x and  y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ( ph  /\ 
 -.  A. x  x  =  y )  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x A. y ps )
 
Theoremnfexd2 2065 Variation on nfexd 1873 which adds the hypothesis that  x and  y are distinct in the inner subproof. (Contributed by Mario Carneiro, 8-Oct-2016.)
 |- 
 F/ y ph   &    |-  ( ( ph  /\ 
 -.  A. x  x  =  y )  ->  F/ x ps )   =>    |-  ( ph  ->  F/ x E. y ps )
 
Theoremexdistrf 2066 Distribution of existential quantifiers, with a bound-variable hypothesis saying that  y is not free in  ph, but  x can be free in  ph (and there is no distinct variable condition on  x and  y). (Contributed by Mario Carneiro, 20-Mar-2013.) (Proof shortened by Wolf Lammen, 14-May-2018.)
 |-  ( -.  A. x  x  =  y  ->  F/ y ph )   =>    |-  ( E. x E. y ( ph  /\  ps )  ->  E. x ( ph  /\ 
 E. y ps )
 )
 
TheoremexdistrfOLD 2067 Obsolete proof of exdistrf 2066 as of 14-May-2018. (Contributed by Mario Carneiro, 20-Mar-2013.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  F/ y ph )   =>    |-  ( E. x E. y ( ph  /\  ps )  ->  E. x ( ph  /\ 
 E. y ps )
 )
 
Theoremdvelimf 2068 Version of dvelimv 2074 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
 |- 
 F/ x ph   &    |-  F/ z ps   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x ps )
 
TheoremdvelimfOLD 2069 Obsolete proof of dvelimf 2068 as of 21-Apr-2018. (Contributed by NM, 1-Oct-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ x ph   &    |-  F/ z ps   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x ps )
 
Theoremdvelimdf 2070 Deduction form of dvelimf 2068. This version may be useful if we want to avoid ax-17 1626 and use ax-16 2221 instead. (Contributed by NM, 7-Apr-2004.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 11-May-2018.)
 |- 
 F/ x ph   &    |-  F/ z ph   &    |-  ( ph  ->  F/ x ps )   &    |-  ( ph  ->  F/ z ch )   &    |-  ( ph  ->  ( z  =  y  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( -.  A. x  x  =  y 
 ->  F/ x ch )
 )
 
Theoremdvelimh 2071 Version of dvelim 2073 without any variable restrictions. (Contributed by NM, 1-Oct-2002.) (Proof shortened by Wolf Lammen, 11-May-2018.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
TheoremdvelimhOLD 2072 Obsolete proof of dvelimh 2071 as of 4-Mar-2018. (Contributed by NM, 1-Oct-2002.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ps  ->  A. z ps )   &    |-  (
 z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdvelim 2073* 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 2070.

Other variants of this theorem are dvelimh 2071 (with no distinct variable restrictions), dvelimhw 1876 (that avoids ax-12 1950), and dvelimALT 2210 (that avoids ax-10 2217). (Contributed by NM, 23-Nov-1994.)

 |-  ( ph  ->  A. x ph )   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdvelimv 2074* Similar to dvelim 2073 with first hypothesis replaced by distinct variable condition. (Contributed by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)
 |-  ( z  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdvelimnf 2075* Version of dvelim 2073 using "not free" notation. (Contributed by Mario Carneiro, 9-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( z  =  y  ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x ps )
 
Theoremdveeq1OLD 2076* Obsolete proof of dveeq1 2021 as of 25-Feb-2018. (Contributed by NM, 2-Jan-2002.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremdveeq2 2077* Quantifier introduction when one pair of variables is distinct. (Contributed by NM, 2-Jan-2002.) (Revised by NM, 20-Jul-2015.)
 |-  ( -.  A. x  x  =  y  ->  ( z  =  y  ->  A. x  z  =  y ) )
 
Theoremax11v2 2078* Recovery of ax-11o 2218 from ax11v 2172. This proof uses ax-10 2217 and ax-11 1761. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 21-Apr-2018.)
 |-  ( x  =  z 
 ->  ( ph  ->  A. x ( x  =  z  -> 
 ph ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) ) )
 
Theoremax11v2OLD 2079* Obsolete proof of ax11v2 2078 as of 21-Apr-2018. (Contributed by NM, 2-Feb-2007.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  z 
 ->  ( ph  ->  A. x ( x  =  z  -> 
 ph ) ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  (
 ph  ->  A. x ( x  =  y  ->  ph )
 ) ) )
 
Theoremax11a2 2080* Derive ax-11o 2218 from a hypothesis in the form of ax-11 1761. ax-10 2217 and ax-11 1761 are used by the proof, but not ax-10o 2216 or ax-11o 2218. TODO: figure out if this is useful, or if it should be simplified or eliminated. (Contributed by NM, 2-Feb-2007.)
 |-  ( x  =  z 
 ->  ( A. z ph  ->  A. x ( x  =  z  ->  ph )
 ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Theoremax11o 2081 Derivation of set.mm's original ax-11o 2218 from ax-10 2217 and the shorter ax-11 1761 that has replaced it.

Theorem ax11 2232 shows the reverse derivation of ax-11 1761 from ax-11o 2218.

Normally, ax11o 2081 should be used rather than ax-11o 2218, except by theorems specifically studying the latter's properties. (Contributed by NM, 3-Feb-2007.)

 |-  ( -.  A. x  x  =  y  ->  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph ) ) ) )
 
Theoremax11b 2082 A bidirectional version of ax11o 2081. (Contributed by NM, 30-Jun-2006.)
 |-  ( ( -.  A. x  x  =  y  /\  x  =  y
 )  ->  ( ph  <->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremequvini 2083 A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require  z to be distinct from  x and  y (making the proof longer). (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 7-Apr-2018.)
 |-  ( x  =  y 
 ->  E. z ( x  =  z  /\  z  =  y ) )
 
TheoremequviniOLD 2084 Obsolete proof of equvini 2083 as of 7-Apr-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( x  =  y 
 ->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremequveli 2085 A variable elimination law for equality with no distinct variable requirements. (Compare equvini 2083.) (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 15-Apr-2018.)
 |-  ( A. z ( z  =  x  <->  z  =  y
 )  ->  x  =  y )
 
TheoremequveliOLD 2086 Obsolete proof of equveli 2085 as of 15-Apr-2018. (Contributed by NM, 1-Mar-2013.) (Proof shortened by Mario Carneiro, 17-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. z ( z  =  x  <->  z  =  y
 )  ->  x  =  y )
 
Theoremequvin 2087* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y  <->  E. z ( x  =  z  /\  z  =  y ) )
 
Theoremequs45f 2088 Two ways of expressing substitution when  y is not free in  ph. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)
 |- 
 F/ y ph   =>    |-  ( E. x ( x  =  y  /\  ph )  <->  A. x ( x  =  y  ->  ph )
 )
 
Theoremequs5 2089 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremsb2 2090 One direction of a simplified definition of substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  [ y  /  x ] ph )
 
Theoremstdpc4 2091 The specialization axiom of standard predicate calculus. It states that if a statement  ph holds for all  x, then it also holds for the specific case of  y (properly) substituted for  x. Translated to traditional notation, it can be read: " A. x ph ( x )  ->  ph ( y ), provided that  y is free for  x in  ph (
x )." Axiom 4 of [Mendelson] p. 69. See also spsbc 3173 and rspsbc 3239. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x ph  ->  [ y  /  x ] ph )
 
Theoremsb3 2092 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( E. x ( x  =  y  /\  ph )  ->  [ y  /  x ] ph ) )
 
Theoremsb4 2093 One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  ph ) ) )
 
Theoremsb4b 2094 Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  <->  A. x ( x  =  y  ->  ph )
 ) )
 
Theoremhbsb2 2095 Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( -.  A. x  x  =  y  ->  ( [ y  /  x ] ph  ->  A. x [
 y  /  x ] ph ) )
 
Theoremnfsb2 2096 Bound-variable hypothesis builder for substitution. (Contributed by Mario Carneiro, 4-Oct-2016.)
 |-  ( -.  A. x  x  =  y  ->  F/ x [ y  /  x ] ph )
 
Theoremhbsb2a 2097 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] A. y ph  ->  A. x [ y  /  x ] ph )
 
Theoremhbsb2e 2098 Special case of a bound-variable hypothesis builder for substitution. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] E. y ph )
 
Theoremhbsb3 2099 If  y is not free in  ph,  x is not free in  [ y  /  x ] ph. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  A. y ph )   =>    |-  ( [ y  /  x ] ph  ->  A. x [ y  /  x ] ph )
 
Theoremnfs1 2100 If  y is not free in  ph,  x is not free in  [ y  /  x ] ph. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ y ph   =>    |- 
 F/ x [ y  /  x ] ph
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