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Theorem mo2v 2269
Description: Alternate definition of "at most one." Unlike mo2 2274, which is slightly more general, it does not depend on ax-11 1782 and ax-13 1955, whence it is preferable within predicate logic. Elsewhere, most theorems depend on these axioms anyway, so this advantage is no longer important. (Contributed by Wolf Lammen, 27-May-2019.) (Proof shortened by Wolf Lammen, 10-Nov-2019.)
Assertion
Ref Expression
mo2v  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem mo2v
StepHypRef Expression
1 df-mo 2267 . 2  |-  ( E* x ph  <->  ( E. x ph  ->  E! x ph ) )
2 df-eu 2266 . . 3  |-  ( E! x ph  <->  E. y A. x ( ph  <->  x  =  y ) )
32imbi2i 312 . 2  |-  ( ( E. x ph  ->  E! x ph )  <->  ( E. x ph  ->  E. y A. x ( ph  <->  x  =  y ) ) )
4 alnex 1589 . . . . . . 7  |-  ( A. x  -.  ph  <->  -.  E. x ph )
5 pm2.21 108 . . . . . . . 8  |-  ( -. 
ph  ->  ( ph  ->  x  =  y ) )
65alimi 1605 . . . . . . 7  |-  ( A. x  -.  ph  ->  A. x
( ph  ->  x  =  y ) )
74, 6sylbir 213 . . . . . 6  |-  ( -. 
E. x ph  ->  A. x ( ph  ->  x  =  y ) )
87eximi 1626 . . . . 5  |-  ( E. y  -.  E. x ph  ->  E. y A. x
( ph  ->  x  =  y ) )
9819.23bi 1810 . . . 4  |-  ( -. 
E. x ph  ->  E. y A. x (
ph  ->  x  =  y ) )
10 bi1 186 . . . . . 6  |-  ( (
ph 
<->  x  =  y )  ->  ( ph  ->  x  =  y ) )
1110alimi 1605 . . . . 5  |-  ( A. x ( ph  <->  x  =  y )  ->  A. x
( ph  ->  x  =  y ) )
1211eximi 1626 . . . 4  |-  ( E. y A. x (
ph 
<->  x  =  y )  ->  E. y A. x
( ph  ->  x  =  y ) )
139, 12ja 161 . . 3  |-  ( ( E. x ph  ->  E. y A. x (
ph 
<->  x  =  y ) )  ->  E. y A. x ( ph  ->  x  =  y ) )
14 nfia1 1892 . . . . . 6  |-  F/ x
( A. x (
ph  ->  x  =  y )  ->  A. x
( ph  <->  x  =  y
) )
15 id 22 . . . . . . . . . 10  |-  ( ph  ->  ph )
16 ax12v 1795 . . . . . . . . . . 11  |-  ( x  =  y  ->  ( ph  ->  A. x ( x  =  y  ->  ph )
) )
1716com12 31 . . . . . . . . . 10  |-  ( ph  ->  ( x  =  y  ->  A. x ( x  =  y  ->  ph )
) )
1815, 17embantd 54 . . . . . . . . 9  |-  ( ph  ->  ( ( ph  ->  x  =  y )  ->  A. x ( x  =  y  ->  ph ) ) )
1918spsd 1807 . . . . . . . 8  |-  ( ph  ->  ( A. x (
ph  ->  x  =  y )  ->  A. x
( x  =  y  ->  ph ) ) )
2019ancld 553 . . . . . . 7  |-  ( ph  ->  ( A. x (
ph  ->  x  =  y )  ->  ( A. x ( ph  ->  x  =  y )  /\  A. x ( x  =  y  ->  ph ) ) ) )
21 albiim 1666 . . . . . . 7  |-  ( A. x ( ph  <->  x  =  y )  <->  ( A. x ( ph  ->  x  =  y )  /\  A. x ( x  =  y  ->  ph ) ) )
2220, 21syl6ibr 227 . . . . . 6  |-  ( ph  ->  ( A. x (
ph  ->  x  =  y )  ->  A. x
( ph  <->  x  =  y
) ) )
2314, 22exlimi 1850 . . . . 5  |-  ( E. x ph  ->  ( A. x ( ph  ->  x  =  y )  ->  A. x ( ph  <->  x  =  y ) ) )
2423eximdv 1677 . . . 4  |-  ( E. x ph  ->  ( E. y A. x (
ph  ->  x  =  y )  ->  E. y A. x ( ph  <->  x  =  y ) ) )
2524com12 31 . . 3  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  ( E. x ph  ->  E. y A. x ( ph  <->  x  =  y ) ) )
2613, 25impbii 188 . 2  |-  ( ( E. x ph  ->  E. y A. x (
ph 
<->  x  =  y ) )  <->  E. y A. x
( ph  ->  x  =  y ) )
271, 3, 263bitri 271 1  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1368   E.wex 1587   E!weu 2262   E*wmo 2263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-eu 2266  df-mo 2267
This theorem is referenced by:  mo2  2274  eu3v  2294  mo3  2307  mo3OLD  2308  sbmo  2324  moim  2328  mopick  2348  2mo2  2368  mo2icl  3245  moabex  4662  dffun3  5540  dffun6f  5543  grothprim  9115  wl-mo2dnae  28563  wl-mo2t  28564  wl-mo3t  28565
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