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Theorem mo2 2276
Description: Alternate definition of "at most one." (Contributed by NM, 8-Mar-1995.) Restrict dummy variable z. (Revised by Wolf Lammen, 28-May-2019.)
Hypothesis
Ref Expression
mo2.1  |-  F/ y
ph
Assertion
Ref Expression
mo2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)

Proof of Theorem mo2
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 mo2v 2273 . 2  |-  ( E* x ph  <->  E. z A. x ( ph  ->  x  =  z ) )
2 mo2.1 . . . . 5  |-  F/ y
ph
3 nfv 1752 . . . . 5  |-  F/ y  x  =  z
42, 3nfim 1977 . . . 4  |-  F/ y ( ph  ->  x  =  z )
54nfal 2004 . . 3  |-  F/ y A. x ( ph  ->  x  =  z )
6 nfv 1752 . . 3  |-  F/ z A. x ( ph  ->  x  =  y )
7 equequ2 1850 . . . . 5  |-  ( z  =  y  ->  (
x  =  z  <->  x  =  y ) )
87imbi2d 318 . . . 4  |-  ( z  =  y  ->  (
( ph  ->  x  =  z )  <->  ( ph  ->  x  =  y ) ) )
98albidv 1758 . . 3  |-  ( z  =  y  ->  ( A. x ( ph  ->  x  =  z )  <->  A. x
( ph  ->  x  =  y ) ) )
105, 6, 9cbvex 2077 . 2  |-  ( E. z A. x (
ph  ->  x  =  z )  <->  E. y A. x
( ph  ->  x  =  y ) )
111, 10bitri 253 1  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188   A.wal 1436   E.wex 1660   F/wnf 1664   E*wmo 2267
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1661  df-nf 1665  df-eu 2270  df-mo 2271
This theorem is referenced by:  mo3  2304  mo  2305  rmo2  3389  nmo  28113  bj-eu3f  31406  bj-mo3OLD  31411
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