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Theorem List for Metamath Proof Explorer - 2301-2400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremexmoeu2 2301 Existence implies "at most one" is equivalent to uniqueness. (Contributed by NM, 5-Apr-2004.)
 |-  ( E. x ph  ->  ( E* x ph  <->  E! x ph ) )
 
Theoremeu3v 2302* An alternate way to express existential uniqueness. (Contributed by NM, 8-Jul-1994.) Add a distinct variable condition on  ph. (Revised by Wolf Lammen, 29-May-2019.)
 |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
 
Theoremeumo 2303 Existential uniqueness implies "at most one." (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x ph  ->  E* x ph )
 
Theoremeumoi 2304 "At most one" inferred from existential uniqueness. (Contributed by NM, 5-Apr-1995.)
 |- 
 E! x ph   =>    |- 
 E* x ph
 
Theoremmoabs 2305 Absorption of existence condition by "at most one." (Contributed by NM, 4-Nov-2002.)
 |-  ( E* x ph  <->  ( E. x ph  ->  E* x ph ) )
 
Theoremexmoeu 2306 Existence in terms of "at most one" and uniqueness. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Wolf Lammen, 5-Dec-2018.)
 |-  ( E. x ph  <->  ( E* x ph  ->  E! x ph ) )
 
TheoremexmoeuOLD 2307 Obsolete proof of exmoeu 2306 as of 5-Dec-2018. (Contributed by NM, 5-Apr-2004.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E. x ph  <->  ( E* x ph  ->  E! x ph ) )
 
Theoremeumo0OLD 2308* Obsolete as of 6-Jun-2018. Existential uniqueness implies "at most one." (Contributed by NM, 8-Jul-1994.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  ->  E. y A. x ( ph  ->  x  =  y ) )
 
Theoremsb8eu 2309 Variable substitution in uniqueness quantifier. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 24-Aug-2019.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
 
Theoremsb8euOLD 2310 Obsolete proof of sb8eu 2309 as of 24-Aug-2019. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 8-Oct-2018.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
 
Theoremsb8euOLDOLD 2311 Obsolete proof of sb8eu 2309 as of 8-Oct-2018. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  E! y [ y  /  x ] ph )
 
Theoremsb8mo 2312 Variable substitution for "at most one." (Contributed by Alexander van der Vekens, 17-Jun-2017.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  E* y [ y  /  x ] ph )
 
Theoremcbveu 2313 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 25-Nov-1994.) (Revised by Mario Carneiro, 7-Oct-2016.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E! x ph  <->  E! y ps )
 
Theoremcbvmo 2314 Rule used to change bound variables, using implicit substitution. (Contributed by NM, 9-Mar-1995.) (Revised by Andrew Salmon, 8-Jun-2011.)
 |- 
 F/ y ph   &    |-  F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ph  <->  E* y ps )
 
Theoremmo3 2315* Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that  y not occur in  ph in place of our hypothesis. (Contributed by NM, 8-Mar-1995.) (Proof shortened by Wolf Lammen, 18-Aug-2019.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
 
Theoremmo3OLD 2316* Obsolete proof of mo3 2315 as of 20-Jul-2019. (Contributed by NM, 8-Mar-1995.) (Revised by Wolf Lammen, 3-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
 
Theoremmo 2317* Equivalent definitions of "there exists at most one." (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
 |- 
 F/ y ph   =>    |-  ( E. y A. x ( ph  ->  x  =  y )  <->  A. x A. y
 ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) )
 
Theoremmo3OLDOLD 2318* Obsolete proof of mo3 2315 as of 2-Dec-2018. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) )
 
Theoremeu2 2319* An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26. (Contributed by NM, 8-Jul-1994.) (Proof shortened by Wolf Lammen, 2-Dec-2018.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
 
Theoremeu1 2320* An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 29-Oct-2018.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) ) )
 
Theoremeu1OLD 2321* Obsolete proof of eu1 2320 as of 29-Oct-2018. (Contributed by NM, 20-Aug-1993.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) ) )
 
TheoremmoOLD 2322* Obsolete proof of mo 2317 as of 7-Oct-2018. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y ph   =>    |-  ( E. y A. x ( ph  ->  x  =  y )  <->  A. x A. y
 ( ( ph  /\  [
 y  /  x ] ph )  ->  x  =  y ) )
 
TheoremeuexALT 2323 Alternate proof of euex 2298. Shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E! x ph  ->  E. x ph )
 
Theoremeu2OLD 2324* Obsolete proof of eu2 2319 as of 2-Dec-2018. (Contributed by NM, 8-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( (
 ph  /\  [ y  /  x ] ph )  ->  x  =  y ) ) )
 
Theoremeu3OLD 2325* Obsolete theorem as of 29-May-2018. Superseded by eu3v 2302 that better fits common usage pattern. (Contributed by NM, 8-Jul-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y ph   =>    |-  ( E! x ph  <->  ( E. x ph  /\  E. y A. x ( ph  ->  x  =  y ) ) )
 
Theoremeu5OLD 2326 Obsolete proof of eu5 2300 as of 25-May-2019 (Contributed by NM, 23-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E! x ph  <->  ( E. x ph  /\  E* x ph ) )
 
Theoremeuor 2327 Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.)
 |- 
 F/ x ph   =>    |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremeuorv 2328* Introduce a disjunct into a uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( ( -.  ph  /\ 
 E! x ps )  ->  E! x ( ph  \/  ps ) )
 
Theoremeuor2 2329 Introduce or eliminate a disjunct in a uniqueness quantifier. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
 |-  ( -.  E. x ph 
 ->  ( E! x (
 ph  \/  ps )  <->  E! x ps ) )
 
Theoremeuor2OLD 2330 Obsolete proof of euor2 2329 as of 27-Dec-2018. (Contributed by NM, 21-Oct-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( -.  E. x ph 
 ->  ( E! x (
 ph  \/  ps )  <->  E! x ps ) )
 
Theoremmo2OLD 2331* Obsolete proof of mo2 2282 as of 28-May-2019. (Contributed by NM, 8-Mar-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y ph   =>    |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
 
Theoremsbmo 2332* Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( [ y  /  x ] E* z ph  <->  E* z [ y  /  x ] ph )
 
Theoremmo4f 2333* "At most one" expressed using implicit substitution. (Contributed by NM, 10-Apr-2004.)
 |- 
 F/ x ps   &    |-  ( x  =  y  ->  (
 ph 
 <->  ps ) )   =>    |-  ( E* x ph  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremmo4 2334* "At most one" expressed using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E* x ph  <->  A. x A. y ( ( ph  /\  ps )  ->  x  =  y ) )
 
Theoremeu4 2335* Uniqueness using implicit substitution. (Contributed by NM, 26-Jul-1995.)
 |-  ( x  =  y 
 ->  ( ph  <->  ps ) )   =>    |-  ( E! x ph  <->  ( E. x ph  /\  A. x A. y ( (
 ph  /\  ps )  ->  x  =  y ) ) )
 
Theoremmoim 2336 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E* x ps  ->  E* x ph ) )
 
Theoremmoimi 2337 "At most one" is preserved through implication (notice wff reversal). (Contributed by NM, 15-Feb-2006.)
 |-  ( ph  ->  ps )   =>    |-  ( E* x ps  ->  E* x ph )
 
TheoremmorimOLD 2338 Obsolete theorem as of 22-Dec-2018. Use moimi 2337 applied to ax-1 6, as demonstrated in the proof. (Contributed by NM, 28-Jul-1995.) (Proof shortened by Wolf Lammen, 22-Dec-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E* x (
 ph  ->  ps )  ->  ( ph  ->  E* x ps )
 )
 
TheoremmorimvOLD 2339* Obsolete proof of morimOLD 2338 as of 22-Dec-2018. (Contributed by NM, 28-Jul-1995.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E* x (
 ph  ->  ps )  ->  ( ph  ->  E* x ps )
 )
 
Theoremeuimmo 2340 Uniqueness implies "at most one" through implication. (Contributed by NM, 22-Apr-1995.)
 |-  ( A. x (
 ph  ->  ps )  ->  ( E! x ps  ->  E* x ph ) )
 
Theoremeuim 2341 Add existential uniqueness quantifiers to an implication. Note the reversed implication in the antecedent. (Contributed by NM, 19-Oct-2005.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)
 |-  ( ( E. x ph 
 /\  A. x ( ph  ->  ps ) )  ->  ( E! x ps  ->  E! x ph ) )
 
Theoremmoan 2342 "At most one" is still the case when a conjunct is added. (Contributed by NM, 22-Apr-1995.)
 |-  ( E* x ph  ->  E* x ( ps 
 /\  ph ) )
 
Theoremmoani 2343 "At most one" is still true when a conjunct is added. (Contributed by NM, 9-Mar-1995.)
 |- 
 E* x ph   =>    |- 
 E* x ( ps 
 /\  ph )
 
Theoremmoor 2344 "At most one" is still the case when a disjunct is removed. (Contributed by NM, 5-Apr-2004.)
 |-  ( E* x (
 ph  \/  ps )  ->  E* x ph )
 
Theoremmooran1 2345 "At most one" imports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E* x ph 
 \/  E* x ps )  ->  E* x ( ph  /\ 
 ps ) )
 
Theoremmooran2 2346 "At most one" exports disjunction to conjunction. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E* x (
 ph  \/  ps )  ->  ( E* x ph  /\ 
 E* x ps )
 )
 
Theoremmoanim 2347 Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
 |- 
 F/ x ph   =>    |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremeuan 2348 Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
 |- 
 F/ x ph   =>    |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
TheoremeuanOLD 2349 Obsolete poof of euan 2348 as of 24-Dec-2018. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 F/ x ph   =>    |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
TheoremmoanimOLD 2350 Obsolete proof of moanim 2347 as of 23-Dec-2018. (Contributed by NM, 3-Dec-2001.) (New usage is discouraged.) (Proof modification is discouraged.)
 |- 
 F/ x ph   =>    |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremmoanimv 2351* Introduction of a conjunct into "at most one" quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E* x (
 ph  /\  ps )  <->  (
 ph  ->  E* x ps )
 )
 
Theoremmoanmo 2352 Nested "at most one" quantifiers. (Contributed by NM, 25-Jan-2006.)
 |- 
 E* x ( ph  /\ 
 E* x ph )
 
Theoremmoaneu 2353 Nested "at most one" and uniqueness quantifiers. (Contributed by NM, 25-Jan-2006.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
 |- 
 E* x ( ph  /\ 
 E! x ph )
 
TheoremmoaneuOLD 2354 Obsolete proof of moaneu 2353 as of 27-Dec-2018. (Contributed by NM, 25-Jan-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 E* x ( ph  /\ 
 E! x ph )
 
Theoremeuanv 2355* Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 23-Mar-1995.)
 |-  ( E! x (
 ph  /\  ps )  <->  (
 ph  /\  E! x ps ) )
 
Theoremmopick 2356 "At most one" picks a variable value, eliminating an existential quantifier. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 17-Sep-2019.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
TheoremmopickOLD 2357 Obsolete proof of mopick 2356 as of 15-Sep-2019. (Contributed by NM, 27-Jan-1997.) (Proof shortened by Wolf Lammen, 29-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
TheoremmopickOLDOLD 2358 Obsolete proof of mopick 2356 as of 27-Dec-2018. (Contributed by NM, 27-Jan-1997.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremeupick 2359 Existential uniqueness "picks" a variable value for which another wff is true. If there is only one thing  x such that 
ph is true, and there is also an  x (actually the same one) such that  ph and  ps are both true, then  ph implies  ps regardless of  x. This theorem can be useful for eliminating existential quantifiers in a hypothesis. Compare Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by NM, 10-Jul-1994.)
 |-  ( ( E! x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  ( ph  ->  ps )
 )
 
Theoremeupicka 2360 Version of eupick 2359 with closed formulas. (Contributed by NM, 6-Sep-2008.)
 |-  ( ( E! x ph 
 /\  E. x ( ph  /\ 
 ps ) )  ->  A. x ( ph  ->  ps ) )
 
Theoremeupickb 2361 Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
 |-  ( ( E! x ph 
 /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  <->  ps ) )
 
TheoremeupickbOLD 2362 Obsolete proof of eupickb 2361 as of 27-Dec-2018. (Contributed by NM, 25-Nov-1994.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( E! x ph 
 /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  <->  ps ) )
 
Theoremeupickbi 2363 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof shortened by Wolf Lammen, 27-Dec-2018.)
 |-  ( E! x ph  ->  ( E. x (
 ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
 
TheoremeupickbiOLD 2364 Theorem *14.26 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( E! x ph  ->  ( E. x (
 ph  /\  ps )  <->  A. x ( ph  ->  ps ) ) )
 
Theoremmopick2 2365 "At most one" can show the existence of a common value. In this case we can infer existence of conjunction from a conjunction of existence, and it is one way to achieve the converse of 19.40 1651. (Contributed by NM, 5-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E* x ph 
 /\  E. x ( ph  /\ 
 ps )  /\  E. x ( ph  /\  ch ) )  ->  E. x ( ph  /\  ps  /\  ch ) )
 
Theoremmoexex 2366 "At most one" double quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 28-Dec-2018.)
 |- 
 F/ y ph   =>    |-  ( ( E* x ph 
 /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
 
TheoremmoexexOLD 2367 Obsolete proof of moexex 2366 as of 28-Dec.2018. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 F/ y ph   =>    |-  ( ( E* x ph 
 /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
 
Theoremmoexexv 2368* "At most one" double quantification. (Contributed by NM, 26-Jan-1997.)
 |-  ( ( E* x ph 
 /\  A. x E* y ps )  ->  E* y E. x ( ph  /\  ps ) )
 
Theorem2moex 2369 Double quantification with "at most one." (Contributed by NM, 3-Dec-2001.)
 |-  ( E* x E. y ph  ->  A. y E* x ph )
 
Theorem2euex 2370 Double quantification with existential uniqueness. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( E! x E. y ph  ->  E. y E! x ph )
 
Theorem2eumo 2371 Double quantification with existential uniqueness and "at most one." (Contributed by NM, 3-Dec-2001.)
 |-  ( E! x E* y ph  ->  E* x E! y ph )
 
Theorem2eu2ex 2372 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( E! x E! y ph  ->  E. x E. y ph )
 
Theorem2moswap 2373 A condition allowing swap of "at most one" and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
 |-  ( A. x E* y ph  ->  ( E* x E. y ph  ->  E* y E. x ph ) )
 
Theorem2euswap 2374 A condition allowing swap of uniqueness and existential quantifiers. (Contributed by NM, 10-Apr-2004.)
 |-  ( A. x E* y ph  ->  ( E! x E. y ph  ->  E! y E. x ph ) )
 
Theorem2exeu 2375 Double existential uniqueness implies double uniqueness quantification. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Mario Carneiro, 22-Dec-2016.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  ->  E! x E! y ph )
 
Theorem2mo2 2376* This theorem extends the idea of "at most one" to expressions in two set variables ("at most one pair  x and  y". Note: this is not expressed by  E* x E* y ph). 2eu4 2385 relates this extension to double existential uniqueness, if at least one pair exists. (Contributed by Wolf Lammen, 26-Oct-2019.)
 |-  ( ( E* x E. y ph  /\  E* y E. x ph )  <->  E. z E. w A. x A. y ( ph  ->  ( x  =  z 
 /\  y  =  w ) ) )
 
Theorem2mo 2377* Two equivalent expressions for double "at most one." (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Nov-2019.)
 |-  ( E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\  [
 z  /  x ] [ w  /  y ] ph )  ->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2moOLD 2378* Obsolete proof of 2mo 2377 as of 2-Nov-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 25-Oct-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\  [
 z  /  x ] [ w  /  y ] ph )  ->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2moOLDOLD 2379* Obsolete proof of 2mo 2377 as of 19-Jan-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) )  <->  A. x A. y A. z A. w ( ( ph  /\  [
 z  /  x ] [ w  /  y ] ph )  ->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2mos 2380* Double "exists at most one", using implicit substitution. (Contributed by NM, 10-Feb-2005.)
 |-  ( ( x  =  z  /\  y  =  w )  ->  ( ph 
 <->  ps ) )   =>    |-  ( E. z E. w A. x A. y ( ph  ->  ( x  =  z  /\  y  =  w )
 ) 
 <-> 
 A. x A. y A. z A. w ( ( ph  /\  ps )  ->  ( x  =  z  /\  y  =  w ) ) )
 
Theorem2eu1 2381 Double existential uniqueness. This theorem shows a condition under which a "naive" definition matches the correct one. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 11-Nov-2019.)
 |-  ( A. x E* y ph  ->  ( E! x E! y ph  <->  ( E! x E. y ph  /\  E! y E. x ph )
 ) )
 
Theorem2eu1OLD 2382 Obsolete proof of 2eu1 2381 as of 11-Nov-2019. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( A. x E* y ph  ->  ( E! x E! y ph  <->  ( E! x E. y ph  /\  E! y E. x ph )
 ) )
 
Theorem2eu2 2383 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( E! y E. x ph  ->  ( E! x E! y ph  <->  E! x E. y ph ) )
 
Theorem2eu3 2384 Double existential uniqueness. (Contributed by NM, 3-Dec-2001.)
 |-  ( A. x A. y ( E* x ph 
 \/  E* y ph )  ->  ( ( E! x E! y ph  /\  E! y E! x ph )  <->  ( E! x E. y ph  /\  E! y E. x ph ) ) )
 
Theorem2eu4 2385* This theorem provides us with a definition of double existential uniqueness ("exactly one 
x and exactly one  y"). Naively one might think (incorrectly) that it could be defined by  E! x E! y ph. See 2eu1 2381 for a condition under which the naive definition holds and 2exeu 2375 for a one-way implication. See 2eu5 2387 and 2eu8 2391 for alternate definitions. (Contributed by NM, 3-Dec-2001.) (Proof shortened by Wolf Lammen, 14-Sep-2019.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
 
Theorem2eu4OLD 2386* Obsolete proof of 2eu4 2385 as of 14-Sep-2019. (Contributed by NM, 3-Dec-2001.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
 
Theorem2eu5 2387* An alternate definition of double existential uniqueness (see 2eu4 2385). A mistake sometimes made in the literature is to use  E! x E! y to mean "exactly one  x and exactly one  y." (For example, see Proposition 7.53 of [TakeutiZaring] p. 53.) It turns out that this is actually a weaker assertion, as can be seen by expanding out the formal definitions. This theorem shows that the erroneous definition can be repaired by conjoining 
A. x E* y ph as an additional condition. The correct definition apparently has never been published. ( E* means "exists at most one.") (Contributed by NM, 26-Oct-2003.)
 |-  ( ( E! x E! y ph  /\  A. x E* y ph )  <->  ( E. x E. y ph  /\  E. z E. w A. x A. y
 ( ph  ->  ( x  =  z  /\  y  =  w ) ) ) )
 
Theorem2eu6 2388* Two equivalent expressions for double existential uniqueness. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof shortened by Wolf Lammen, 2-Oct-2019.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E. z E. w A. x A. y ( ph  <->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2eu6OLD 2389* Obsolete proof of 2eu6 2388 as of 21-Sep-2019. (Contributed by NM, 2-Feb-2005.) (Revised by Mario Carneiro, 17-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E. z E. w A. x A. y ( ph  <->  ( x  =  z  /\  y  =  w )
 ) )
 
Theorem2eu7 2390 Two equivalent expressions for double existential uniqueness. (Contributed by NM, 19-Feb-2005.)
 |-  ( ( E! x E. y ph  /\  E! y E. x ph )  <->  E! x E! y ( E. x ph  /\  E. y ph ) )
 
Theorem2eu8 2391 Two equivalent expressions for double existential uniqueness. Curiously, we can put  E! on either of the internal conjuncts but not both. We can also commute  E! x E! y using 2eu7 2390. (Contributed by NM, 20-Feb-2005.)
 |-  ( E! x E! y ( E. x ph 
 /\  E. y ph )  <->  E! x E! y ( E! x ph  /\  E. y ph ) )
 
Theoremeuequ1OLD 2392* Obsolete proof of euequ1 2276 as of 8-Sep-2019. (Contributed by Stefan Allan, 4-Dec-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
 |- 
 E! x  x  =  y
 
Theoremexists1 2393* Two ways to express "only one thing exists." The left-hand side requires only one variable to express this. Both sides are false in set theory; see theorem dtru 4633. (Contributed by NM, 5-Apr-2004.)
 |-  ( E! x  x  =  x  <->  A. x  x  =  y )
 
Theoremexists2 2394 A condition implying that at least two things exist. (Contributed by NM, 10-Apr-2004.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
 |-  ( ( E. x ph 
 /\  E. x  -.  ph )  ->  -.  E! x  x  =  x )
 
1.8  Other axiomatizations related to classical predicate calculus
 
1.8.1  Predicate calculus with all distinct variables
 
Axiomax-7d 2395* Distinct variable version of ax-11 1786. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( A. x A. y ph  ->  A. y A. x ph )
 
Axiomax-8d 2396* Distinct variable version of ax-7 1734. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( x  =  y 
 ->  ( x  =  z 
 ->  y  =  z
 ) )
 
Axiomax-9d1 2397 Distinct variable version of ax-6 1714, equal variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 -.  A. x  -.  x  =  x
 
Axiomax-9d2 2398* Distinct variable version of ax-6 1714, distinct variables case. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |- 
 -.  A. x  -.  x  =  y
 
Axiomax-10d 2399* Distinct variable version of axc11n 2017. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( A. x  x  =  y  ->  A. y  y  =  x )
 
Axiomax-11d 2400* Distinct variable version of ax-12 1798. (Contributed by Mario Carneiro, 14-Aug-2015.)
 |-  ( x  =  y 
 ->  ( A. y ph  ->  A. x ( x  =  y  ->  ph )
 ) )
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