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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnf3 1801 An alternative definition of df-nf 1532. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  <->  A. x ( E. x ph 
 ->  ph ) )
 
Theoremnf4 1802 Variable  x is effectively not free in  ph iff  ph is always true or always false. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |-  ( F/ x ph  <->  ( A. x ph  \/  A. x  -.  ph ) )
 
Theoremexlimi 1803 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ps   &    |-  ( ph  ->  ps )   =>    |-  ( E. x ph  ->  ps )
 
Theorem19.23bi 1804 Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( E. x ph  ->  ps )   =>    |-  ( ph  ->  ps )
 
Theoremexlimd 1805 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch )
 )
 
Theoremexlimdh 1806 Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jan-1997.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ch  ->  A. x ch )   &    |-  ( ph  ->  ( ps  ->  ch ) )   =>    |-  ( ph  ->  ( E. x ps  ->  ch )
 )
 
Theorem19.27 1807 Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  /\  ps )  <->  (
 A. x ph  /\  ps ) )
 
Theorem19.28 1808 Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  /\  ps )  <->  (
 ph  /\  A. x ps ) )
 
Theorem19.36 1809 Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  ->  ps )  <->  (
 A. x ph  ->  ps ) )
 
Theorem19.36i 1810 Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   &    |-  E. x ( ph  ->  ps )   =>    |-  ( A. x ph  ->  ps )
 
Theorem19.37 1811 Theorem 19.37 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( E. x (
 ph  ->  ps )  <->  ( ph  ->  E. x ps ) )
 
Theorem19.38 1812 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |-  ( ( E. x ph 
 ->  A. x ps )  ->  A. x ( ph  ->  ps ) )
 
Theorem19.32 1813 Theorem 19.32 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   =>    |-  ( A. x (
 ph  \/  ps )  <->  (
 ph  \/  A. x ps ) )
 
Theorem19.31 1814 Theorem 19.31 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( A. x ( ph  \/  ps )  <->  (
 A. x ph  \/  ps ) )
 
Theorem19.44 1815 Theorem 19.44 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  \/  ps )  <->  ( E. x ph  \/  ps ) )
 
Theorem19.45 1816 Theorem 19.45 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( E. x (
 ph  \/  ps )  <->  (
 ph  \/  E. x ps ) )
 
Theorem19.41 1817 Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |- 
 F/ x ps   =>    |-  ( E. x ( ph  /\  ps )  <->  ( E. x ph  /\  ps ) )
 
Theorem19.42 1818 Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 18-Aug-1993.)
 |- 
 F/ x ph   =>    |-  ( E. x (
 ph  /\  ps )  <->  (
 ph  /\  E. x ps ) )
 
Theoremexcom13 1819 Swap 1st and 3rd existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. z E. y E. x ph )
 
Theoremexrot3 1820 Rotate existential quantifiers. (Contributed by NM, 17-Mar-1995.)
 |-  ( E. x E. y E. z ph  <->  E. y E. z E. x ph )
 
Theoremexrot4 1821 Rotate existential quantifiers twice. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y E. z E. w ph  <->  E. z E. w E. x E. y ph )
 
Theoremnexr 1822 Inference from 19.8a 1720. (Contributed by Jeff Hankins, 26-Jul-2009.)
 |- 
 -.  E. x ph   =>    |- 
 -.  ph
 
Theoremnfim1 1823 A closed form of nfim 1771. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 24-Sep-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  F/ x (
 ph  ->  ps )
 
Theoremnfan1 1824 A closed form of nfan 1773. (Contributed by Mario Carneiro, 3-Oct-2016.)
 |- 
 F/ x ph   &    |-  ( ph  ->  F/ x ps )   =>    |-  F/ x (
 ph  /\  ps )
 
Theoremexan 1825 Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
 |-  ( E. x ph  /\ 
 ps )   =>    |- 
 E. x ( ph  /\ 
 ps )
 
Theoremhbnd 1826 Deduction form of bound-variable hypothesis builder hbn 1722. (Contributed by NM, 3-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( -.  ps  ->  A. x  -.  ps ) )
 
Theoremaaan 1827 Rearrange universal quantifiers. (Contributed by NM, 12-Aug-1993.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( A. x A. y (
 ph  /\  ps )  <->  (
 A. x ph  /\  A. y ps ) )
 
Theoremeeor 1828 Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)
 |- 
 F/ y ph   &    |-  F/ x ps   =>    |-  ( E. x E. y (
 ph  \/  ps )  <->  ( E. x ph  \/  E. y ps ) )
 
Theoremqexmid 1829 Quantified "excluded middle." Exercise 9.2a of Boolos, p. 111, Computability and Logic. (Contributed by NM, 10-Dec-2000.)
 |- 
 E. x ( ph  ->  A. x ph )
 
Theoremequs5a 1830 A property related to substitution that unlike equs5 1939 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( E. x ( x  =  y  /\  A. y ph )  ->  A. x ( x  =  y  ->  ph ) )
 
Theoremequs5e 1831 A property related to substitution that unlike equs5 1939 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
 
Theoremexlimdd 1832 Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
 |- 
 F/ x ph   &    |-  F/ x ch   &    |-  ( ph  ->  E. x ps )   &    |-  (
 ( ph  /\  ps )  ->  ch )   =>    |-  ( ph  ->  ch )
 
Theorem19.21v 1833* 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 1793 via the use of distinct variable conditions combined with nfv 1605. 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 2150 derived from df-eu 2148. 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 ) )
 
Theorem19.23v 1834* 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 1835* 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 )
 )
 
Theorempm11.53 1836* 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 )
 )
 
Theorem19.27v 1837* Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  ps ) )
 
Theorem19.28v 1838* Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 ph  /\  A. x ps ) )
 
Theorem19.36v 1839* 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 1840* 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 1841* Special case of 19.12 1736 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 1842* 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 1843* 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 1844* 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 1845* 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 1846* 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 1847* 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 1848* 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 1849* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <-> 
 E. x ( ph  /\ 
 E. y ps )
 )
 
Theorem19.42vv 1850* 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 1851* 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 1852* 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 1853* 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 1854* 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 )
 ) ) )
 
Theoremeean 1855 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 1856* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <->  ( E. x ph  /\ 
 E. y ps )
 )
 
Theoremeeeanv 1857* 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 1858* 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 1859* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremstdpc7 1860 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1651.) Translated to traditional notation, it can be read: " x  =  y  ->  ( ph (
x ,  x )  ->  ph ( x ,  y ) ), provided that  y is free for  x in  ph ( x ,  x
)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
 |-  ( x  =  y 
 ->  ( [ x  /  y ] ph  ->  ph )
 )
 
Theoremsbequ1 1861 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  [ y  /  x ] ph )
 )
 
Theoremsbequ12 1862 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  [ y  /  x ] ph ) )
 
Theoremsbequ12r 1863 An equality theorem for substitution. (Contributed by NM, 6-Oct-2004.) (Proof shortened by Andrew Salmon, 21-Jun-2011.)
 |-  ( x  =  y 
 ->  ( [ x  /  y ] ph  <->  ph ) )
 
Theoremsbequ12a 1864 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
 
Theoremsbid 1865 An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). (Contributed by NM, 5-Aug-1993.)
 |-  ( [ x  /  x ] ph  <->  ph )
 
Theoremsb4a 1866 A version of sb4 1992 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] A. y ph  ->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb4e 1867 One direction of a simplified definition of substitution that unlike sb4 1992 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph )
 )
 
1.5.4  Axiom scheme ax-12 (Quantified Equality)
 
Axiomax-12 1868 Axiom of Quantified Equality. One of the equality and substitution axioms of predicate calculus with equality.

An equivalent way to express this axiom that may be easier to understand is  ( -.  x  =  y  ->  ( -.  x  =  z  -> 
( y  =  z  ->  A. x y  =  z ) ) ) (see ax12b 1656). Recall that in the intended interpretation, our variables are metavariables ranging over the variables of predicate calculus (the object language). In order for the first antecedent  -.  x  =  y to hold,  x and  y must have different values and thus cannot be the same object-language variable. Similarly,  x and  z cannot be the same object-language variable. Therefore,  x will not occur in the wff  y  =  z when the first two antecedents hold, so analogous to ax-17 1603, the conclusion  ( y  =  z  ->  A. x
y  =  z ) follows.

The original version of this axiom was ax-12o 2084 and was replaced with this shorter ax-12 1868 in December 2015. The old axiom is proved from this one as theorem ax12o 1877. Conversely, this axiom is proved from ax-12o 2084 as theorem ax12 2097.

The primary purpose of this axiom is to provide a way to introduce the quantifier  A. x on  y  =  z even when  x and  y are substituted with the same variable. In this case, the first antecedent becomes  -.  x  =  x and the axiom still holds.

Although this version is shorter, the original version ax12o 1877 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of ax12o 1877 is in dvelimfALT 1907 which converts a distinct variable pair to the distinctor antecendent  -.  A. x x  =  y.

This axiom can be weakened if desired by adding distinct variable restrictions on pairs  x ,  z and  y ,  z. To show that, we add these restrictions to theorem ax12v 1869 and use only ax12v 1869 for further derivations. Thus ax12v 1869 should be the only theorem referencing this axiom. Other theorems can reference either ax12v 1869 or ax12o 1877.

This axiom scheme is logically redundant (see ax12w 1699) but is used as an auxiliary axiom to achieve metalogical completeness. (Contributed by NM, 21-Dec-2015.) (New usage is discouraged.)

 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax12v 1869* A weaker version of ax-12 1868 with distinct variable restrictions on pairs  x ,  z and  y ,  z. In order to show that this weakening is adequate, this should be the only theorem referencing ax-12 1868 directly. (Contributed by NM, 30-Jun-2016.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theoremax12olem1 1870* Lemma for ax12o 1877. Similar to equvin 1946 but with a negated equality. (Contributed by NM, 24-Dec-2015.)
 |-  ( E. w ( y  =  w  /\  -.  z  =  w )  <->  -.  y  =  z
 )
 
Theoremax12olem2 1871* Lemma for ax12o 1877. Negate the equalities in ax-12 1868, shown as the hypothesis. (Contributed by NM, 24-Dec-2015.)
 |-  ( -.  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w )
 )   =>    |-  ( -.  x  =  y  ->  ( -.  y  =  z  ->  A. x  -.  y  =  z ) )
 
Theoremax12olem3 1872 Lemma for ax12o 1877. Show the equivalence of an intermediate equivalent to ax-12o 2084 with the conjunction of ax-12 1868 and a variant with negated equalities. (Contributed by NM, 24-Dec-2015.)
 |-  ( ( -.  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 ) ) ) )
 
Theoremax12olem4 1873* Lemma for ax12o 1877. Construct an intermediate equivalent to ax-12 1868 from two instances of ax-12 1868. (Contributed by NM, 24-Dec-2015.)
 |-  ( -.  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 ) )
 
Theoremax12olem5 1874 Lemma for ax12o 1877. See ax12olem6 1875 for derivation of ax-12o 2084 from the conclusion. (Contributed by NM, 24-Dec-2015.)
 |-  ( -.  x  =  y  ->  ( -.  A. x  -.  y  =  z  ->  A. x  y  =  z ) )   =>    |-  ( -.  A. x  x  =  y  ->  (
 y  =  z  ->  A. x  y  =  z ) )
 
Theoremax12olem6 1875* Lemma for ax12o 1877. Derivation of ax-12o 2084 from the hypotheses, without using ax-12o 2084. (Contributed by Andrew Salmon, 21-Jul-2011.) (Revised by NM, 24-Dec-2015.)
 |-  ( -.  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 ) ) )
 
Theoremax12olem7 1876* Lemma for ax12o 1877. Derivation of ax-12o 2084 from the hypotheses, without using ax-12o 2084. (Contributed by NM, 24-Dec-2015.)
 |-  ( -.  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 ) ) )
 
Theoremax12o 1877 Derive set.mm's original ax-12o 2084 from the shorter ax-12 1868. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.)
 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theoremax10lem1 1878* Lemma for ax10 1886. Change bound variable. (Contributed by NM, 22-Jul-2015.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  w )
 
Theoremax10lem2 1879* Lemma for ax10 1886. Change free variable. (Contributed by NM, 25-Jul-2015.)
 |-  ( A. x  x  =  y  ->  A. x  x  =  z )
 
Theoremax10lem3 1880* Lemma for ax10 1886. Similar to ax-10 2082 but with distinct variables. (Contributed by NM, 25-Jul-2015.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theoremdvelimv 1881* Similar to dvelim 1961 with first hypothesis replaced by distinct variable condition. (Contributed by NM, 25-Jul-2015.)
 |-  ( z  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( -.  A. x  x  =  y  ->  ( ps  ->  A. x ps ) )
 
Theoremdveeq2 1882* 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 ) )
 
Theoremax10lem4 1883* Lemma for ax10 1886. Change bound variable. (Contributed by NM, 8-Jul-2016.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  x )
 
Theoremax10lem5 1884* Lemma for ax10 1886. Change free and bound variables. (Contributed by NM, 22-Jul-2015.)
 |-  ( A. z  z  =  w  ->  A. y  y  =  x )
 
Theoremax10lem6 1885 Lemma for ax10 1886. Similar to ax10o 1894 but with reversed antecedent. (Contributed by NM, 25-Jul-2015.)
 |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph ) )
 
Theoremax10 1886 Derive set.mm's original ax-10 2082 from others. (Contributed by NM, 25-Jul-2015.) (Revised by NM, 7-Nov-2015.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Theorema16gALT 1887* Alternate proof of a16g 2048 without using ax-4 2077, ax-9 1636, or ax-10 2082 but allowing ax9v 1637. (Contributed by NM, 25-Jul-2015.)
 |-  ( A. x  x  =  y  ->  ( ph  ->  A. z ph )
 )
 
Theoremalequcom 1888 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 )
 
Theoremalequcoms 1889 A commutation rule for identical variable specifiers. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  ph )   =>    |-  ( A. y  y  =  x  ->  ph )
 
Theoremnalequcoms 1890 A commutation rule for distinct variable specifiers. (Contributed by NM, 2-Jan-2002.)
 |-  ( -.  A. x  x  =  y  ->  ph )   =>    |-  ( -.  A. y  y  =  x  ->  ph )
 
Theoremax9 1891 Theorem showing that ax-9 1636 follows from the weaker version ax9v 1637. (Even though this theorem depends on ax-9 1636, all references of ax-9 1636 are made via ax9v 1637. An earlier version stated ax9v 1637 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1636 so that all proofs can be traced back to ax9v 1637. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.)

 |- 
 -.  A. x  -.  x  =  y
 
Theoremax9o 1892 Show that the original axiom ax-9o 2080 can be derived from ax9 1891 and others. See ax9from9o 2089 for the rederivation of ax9 1891 from ax-9o 2080.

Normally, ax9o 1892 should be used rather than ax-9o 2080, except by theorems specifically studying the latter's properties. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.)

 |-  ( A. x ( x  =  y  ->  A. x ph )  ->  ph )
 
Theorema9e 1893 At least one individual exists. This is not a theorem of free logic, which is sound in empty domains. For such a logic, we would add this theorem as an axiom of set theory (Axiom 0 of [Kunen] p. 10). In the system consisting of ax-5 1544 through ax-14 1689 and ax-17 1603, all axioms other than ax9 1891 are believed to be theorems of free logic, although the system without ax9 1891 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.)
 |- 
 E. x  x  =  y
 
Theoremax10o 1894 Show that ax-10o 2081 can be derived from ax-10 2082 in the form of ax10 1886. Normally, ax10o 1894 should be used rather than ax-10o 2081, except by theorems specifically studying the latter's properties. (Contributed by NM, 16-May-2008.) (Proof modification is discouraged.)
 |-  ( A. x  x  =  y  ->  ( A. x ph  ->  A. y ph ) )
 
Theoremhbae 1895 All variables are effectively bound in an identical variable specifier. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. x  x  =  y  ->  A. z A. x  x  =  y )
 
Theoremnfae 1896 All variables are effectively bound in an identical variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z A. x  x  =  y
 
Theoremhbaes 1897 Rule that applies hbae 1895 to antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. z A. x  x  =  y  -> 
 ph )   =>    |-  ( A. x  x  =  y  ->  ph )
 
Theoremhbnae 1898 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 1899 All variables are effectively bound in an distinct variable specifier. (Contributed by Mario Carneiro, 11-Aug-2016.)
 |- 
 F/ z  -.  A. x  x  =  y
 
Theoremhbnaes 1900 Rule that applies hbnae 1898 to antecedent. (Contributed by NM, 5-Aug-1993.)
 |-  ( A. z  -.  A. x  x  =  y 
 ->  ph )   =>    |-  ( -.  A. x  x  =  y  ->  ph )
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