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Theorem List for Metamath Proof Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremhbnd 1901 Deduction form of bound-variable hypothesis builder hbn 1797. (Contributed by NM, 3-Jan-2002.)
 |-  ( ph  ->  A. x ph )   &    |-  ( ph  ->  ( ps  ->  A. x ps ) )   =>    |-  ( ph  ->  ( -.  ps  ->  A. x  -.  ps ) )
 
Theoremaaan 1902 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 1903 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 1904 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 1905 A property related to substitution that unlike equs5 2049 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 1906 A property related to substitution that unlike equs5 2049 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.)
 |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
 
Theoremequs5eOLD 1907 Obsolete proof of equs5e 1906 as of 15-Jan-2018. (Contributed by NM, 2-Feb-2007.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
 
Theoremexlimdd 1908 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 1909* 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 1810 via the use of distinct variable conditions combined with nfv 1626. 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 2260 derived from df-eu 2258. 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 1910* 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 1911* 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 1912* 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 1913* Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 A. x ph  /\  ps ) )
 
Theorem19.28v 1914* Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
 |-  ( A. x (
 ph  /\  ps )  <->  (
 ph  /\  A. x ps ) )
 
Theorem19.36v 1915* 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 1916* 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 1917* Special case of 19.12 1865 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 1918* 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 1919* 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 1920* 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 1921* 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 1922* 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 1923* 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 1924* 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 1925* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <-> 
 E. x ( ph  /\ 
 E. y ps )
 )
 
Theorem19.42vv 1926* 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 1927* 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 1928* 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 1929* 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 1930* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)
 |-  ( E. x E. y E. z E. w ( ( ph  /\  ps )  /\  ( ch  /\  th ) )  <->  E. x ( ph  /\ 
 E. y ( ps 
 /\  E. z ( ch 
 /\  E. w th )
 ) ) )
 
Theorem4exdistrOLD 1931* Obsolete proof of 4exdistr 1930 as of 20-Jan-2018. (Contributed by NM, 9-Mar-1995.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( 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 1932 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 1933* Rearrange existential quantifiers. (Contributed by NM, 26-Jul-1995.)
 |-  ( E. x E. y ( ph  /\  ps ) 
 <->  ( E. x ph  /\ 
 E. y ps )
 )
 
Theoremeeeanv 1934* Rearrange existential quantifiers. Revised to loosen distinct variable restrictions. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Revised by Wolf Lammen, 20-Jan-2018.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps  /\  ch )  <->  ( E. x ph  /\  E. y ps  /\  E. z ch ) )
 
TheoremeeeanvOLD 1935* Obsolete proof of eeeanv 1934 as of 20-Jan-2018. (Contributed by NM, 26-Jul-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E. x E. y E. z ( ph  /\ 
 ps  /\  ch )  <->  ( E. x ph  /\  E. y ps  /\  E. z ch ) )
 
Theoremee4anv 1936* 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 1937* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.)
 |-  ( ph  ->  -.  ps )   =>    |-  ( ph  ->  -.  E. x ps )
 
Theoremstdpc7 1938 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1695.) 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 1939 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  [ y  /  x ] ph )
 )
 
Theoremsbequ12 1940 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  [ y  /  x ] ph ) )
 
Theoremsbequ12r 1941 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 1942 An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( [ y  /  x ] ph  <->  [ x  /  y ] ph ) )
 
Theoremsbid 1943 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 1944 A version of sb4 2102 that doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
 |-  ( [ y  /  x ] A. y ph  ->  A. x ( x  =  y  ->  ph )
 )
 
Theoremsb4e 1945 One direction of a simplified definition of substitution that unlike sb4 2102 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 1946 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 1697). 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 1623, the conclusion  ( y  =  z  ->  A. x
y  =  z ) follows.

The original version of this axiom was ax-12o 2192 and was replaced with this shorter ax-12 1946 in December 2015. The old axiom is proved from this one as theorem ax12o 1976. Conversely, this axiom is proved from ax-12o 2192 as theorem ax12 1985.

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 1976 may be more practical to work with because of the "distinctor" form of its antecedents. A typical application of ax12o 1976 is in dvelimh 2015 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 1947 and use only ax12v 1947 for further derivations. Thus, ax12v 1947 should be the only theorem referencing this axiom. Other theorems can reference either ax12v 1947 or ax12o 1976.

This axiom scheme is logically redundant (see ax12w 1735) 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 1947* A weaker version of ax-12 1946 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 1946 directly. (Contributed by NM, 30-Jun-2016.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )
 
Theorema9e 1948 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 1563 through ax-14 1725 and ax-17 1623, all axioms other than ax-9 1662 are believed to be theorems of free logic, although the system without ax-9 1662 is probably not complete in free logic. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 25-Feb-2018.)
 |- 
 E. x  x  =  y
 
Theoremax9 1949 Theorem showing that ax-9 1662 follows from the weaker version ax9v 1663. (Even though this theorem depends on ax-9 1662, all references of ax-9 1662 are made via ax9v 1663. An earlier version stated ax9v 1663 as a separate axiom, but having two axioms caused some confusion.)

This theorem should be referenced in place of ax-9 1662 so that all proofs can be traced back to ax9v 1663. (Contributed by NM, 12-Nov-2013.) (Revised by NM, 25-Jul-2015.) (Proof shortened by Wolf Lammen, 4-Feb-2018.)

 |- 
 -.  A. x  -.  x  =  y
 
Theoremax9o 1950 Show that the original axiom ax-9o 2188 can be derived from ax9 1949 and others. See ax9from9o 2198 for the rederivation of ax9 1949 from ax-9o 2188.

Normally, ax9o 1950 should be used rather than ax-9o 2188, 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 )
 
Theoremequs4 1951 Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph ) )
 
Theoremequs4OLD 1952 Obsolete proof of equs4 1951 as of 5-Feb-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Mario Carneiro, 20-May-2014.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( A. x ( x  =  y  ->  ph )  ->  E. x ( x  =  y  /\  ph ) )
 
Theoremspimt 1953 Closed theorem form of spim 1955. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Feb-2018.)
 |-  ( ( F/ x ps  /\  A. x ( x  =  y  ->  ( ph  ->  ps )
 ) )  ->  ( A. x ph  ->  ps )
 )
 
TheoremspimtOLD 1954 Obsolete proof of spimt 1953 as of 17-Feb-2018. (Contributed by NM, 15-Jan-2008.) (Revised by Mario Carneiro, 17-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ( F/ x ps  /\  A. x ( x  =  y  ->  ( ph  ->  ps )
 ) )  ->  ( A. x ph  ->  ps )
 )
 
Theoremspim 1955 Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1955 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremspimOLD 1956 Obsolete proof of spim 1955 as of 18-Feb-2018. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) (Proof modofication is discouraged.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph  ->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
TheoremspimeOLD 1957 Obsolete proof of spime 1960 as of 17-Feb-2018. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 F/ x ph   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
Theoremspimed 1958 Deduction version of spime 1960. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 19-Feb-2018.)
 |-  ( ch  ->  F/ x ph )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ch  ->  (
 ph  ->  E. x ps )
 )
 
TheoremspimedOLD 1959 Obsolete proof of spimed 1958 as of 19-Feb-2018. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ch  ->  F/ x ph )   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ch  ->  (
 ph  ->  E. x ps )
 )
 
Theoremspime 1960 Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 6-Mar-2018.)
 |- 
 F/ x ph   &    |-  ( x  =  y  ->  ( ph  ->  ps ) )   =>    |-  ( ph  ->  E. x ps )
 
Theoremspimv 1961* A version of spim 1955 with a distinct variable requirement instead of a bound variable hypothesis. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( A. x ph  ->  ps )
 
Theoremspimev 1962* Distinct-variable version of spime 1960. (Contributed by NM, 5-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  ->  ps )
 )   =>    |-  ( ph  ->  E. x ps )
 
Theoremspv 1963* Specialization, using implicit substitution. (Contributed by NM, 30-Aug-1993.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( A. x ph 
 ->  ps )
 
Theoremspei 1964 Inference from existential specialization, using implicit substitution. Revised to remove a distinct variable constraint. (Contributed by NM, 19-Aug-1993.) (Revised by Wolf Lammen, 23-Feb-2018.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ps   =>    |-  E. x ph
 
TheoremspeivOLD 1965* Obsolete proof of spei 1964 as of 23-Feb-2018. (Contributed by NM, 19-Aug-1993.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   &    |-  ps   =>    |-  E. x ph
 
Theoremequsal 1966 A useful equivalence related to substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 5-Feb-2018.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( x  =  y  -> 
 ph )  <->  ps )
 
TheoremequsalOLD 1967 Obsolete proof of equsal 1966 as of 5-Feb-2018. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( A. x ( x  =  y  -> 
 ph )  <->  ps )
 
Theoremequsalh 1968 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 1969 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 1970 Obsolete proof of equsex 1969 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 1971 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 1972* Lemma for ax12o 1976 and dveeq1 1987. The proof of ax12o 1976 bases on ideas from NM, 24-Dec-2015. (Contributed by Wolf Lammen, 8-Feb-2018.)
 |-  ( y  =  z  <->  A. w ( y  =  w  ->  z  =  w ) )
 
Theoremax12olem2 1973* Lemma for ax12o 1976 and dveeq1 1987. (Contributed by Wolf Lammen, 8-Feb-2018.)
 |-  ( -.  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w )
 )   =>    |-  ( -.  x  =  y  ->  ( E. x  y  =  z  ->  y  =  z ) )
 
Theoremax12olem3 1974* Lemma for ax12o 1976 and dveeq1 1987. (Contributed by Wolf Lammen, 30-Jan-2018.)
 |-  ( -.  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z )
 )   &    |-  ( -.  x  =  y  ->  ( y  =  w  ->  A. x  y  =  w )
 )   =>    |-  ( -.  A. x  x  =  y  ->  F/ x  y  =  z )
 
Theoremax12olem4 1975* Lemma for ax12o 1976. (Contributed by Wolf Lammen, 8-Feb-2018.)
 |-  ( ph  ->  F/ x  y  =  w )   &    |-  ( ps  ->  F/ x  z  =  w )   =>    |-  ( ph  ->  ( ps  ->  ( y  =  z  ->  A. x  y  =  z )
 ) )
 
Theoremax12o 1976 Derive set.mm's original ax-12o 2192 from the shorter ax-12 1946. (Contributed by NM, 29-Nov-2015.) (Revised by NM, 24-Dec-2015.) (Revised by Wolf Lammen, 30-Jan-2018.)
 |-  ( -.  A. z  z  =  x  ->  ( -.  A. z  z  =  y  ->  ( x  =  y  ->  A. z  x  =  y ) ) )
 
Theoremax12olem1OLD 1977* Obsolete proof of ax12oOLD 1984 as of 30-Jan-2018. Lemma for ax12oOLD 1984. Similar to equvin 2051 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 1978* Obsolete proof of ax12oOLD 1984 as of 30-Jan-2018. Lemma for ax12oOLD 1984. Negate the equalities in ax-12 1946, 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 1979 Obsolete proof of ax12oOLD 1984 as of 30-Jan-2018. Lemma for ax12oOLD 1984. Show the equivalence of an intermediate equivalent to ax12o 1976 with the conjunction of ax-12 1946 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 1980* Obsolete proof of ax12oOLD 1984 as of 30-Jan-2018. Lemma for ax12oOLD 1984. Construct an intermediate equivalent to ax-12 1946 from two instances of ax-12 1946. (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 1981 Obsolete proof of ax12oOLD 1984 as of 30-Jan-2018. Lemma for ax12oOLD 1984. See ax12olem6OLD 1982 for derivation of ax12oOLD 1984 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 1982* Obsolete proof of ax12oOLD 1984 as of 30-Jan-2018. Lemma for ax12oOLD 1984. Derivation of ax12oOLD 1984 from the hypotheses, without using ax12oOLD 1984. (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 1983* Obsolete proof of ax12oOLD 1984 as of 30-Jan-2018. Lemma for ax12oOLD 1984. Derivation of ax12oOLD 1984 from the hypotheses, without using ax12oOLD 1984. (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 1984 Obsolete proof of ax12oOLD 1984 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 1985 Derive ax-12 1946 from ax12v 1947 via ax12o 1976. This shows that the weakening in ax12v 1947 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 1986 Obsolete proof of ax12 1985 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 1987* Quantifier introduction when one pair of variables is distinct. Revised to be independent of dvelimv 2017. (Contributed by NM, 2-Jan-2002.) (Revised by Wolf Lammen, 27-Feb-2018.)
 |-  ( -.  A. x  x  =  y  ->  ( y  =  z  ->  A. x  y  =  z ) )
 
Theoremax10lem1 1988* Lemma for ax10 1991. Change bound variable. (Contributed by NM, 22-Jul-2015.)
 |-  ( A. x  x  =  w  ->  A. y  y  =  w )
 
Theoremax10lem2 1989* Lemma for ax10 1991. 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 )
 
Theoremax10lem3 1990 Lemma for ax10 1991. Similar to ax10o 2001 but with reversed antecedent. (Contributed by NM, 25-Jul-2015.) (New usage discouraged.)
 |-  ( A. y  y  =  x  ->  ( A. x ph  ->  A. y ph ) )
 
Theoremax10 1991 Derive set.mm's original ax-10 2190 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 1992* Obsolete proof of a lemma for ax10 1991 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 1993* Obsolete proof of a lemma for ax10 1991 as of 17-Feb-2018. Similar to ax-10 2190 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 1994* Obsolete proof of dvelimv 2017 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 1995* Obsolete proof of dveeq2 2019 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 1996* Obsolete proof of ax10lem2 1989 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 1997* Obsolete proof of ax10lem3 1990 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 1998 Obsolete proof of ax10 1991 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 1999 Obsolete proof of ax9 1949 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 2000 Obsolete proof of a9e 1948 as of 4-Feb-2018. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.) (Proof modfication is discouraged.)
 |- 
 E. x  x  =  y
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