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Theorem rmoeq 27801
Description: Equality's restricted existential "at most one" property. (Contributed by Thierry Arnoux, 30-Mar-2018.)
Assertion
Ref Expression
rmoeq  |-  ( A  e.  V  ->  E* x  e.  B  x  =  A )
Distinct variable groups:    x, A    x, B
Allowed substitution hint:    V( x)

Proof of Theorem rmoeq
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 id 22 . . . 4  |-  ( x  =  A  ->  x  =  A )
21rgenw 2765 . . 3  |-  A. x  e.  B  ( x  =  A  ->  x  =  A )
3 eqeq2 2417 . . . . . 6  |-  ( y  =  A  ->  (
x  =  y  <->  x  =  A ) )
43imbi2d 314 . . . . 5  |-  ( y  =  A  ->  (
( x  =  A  ->  x  =  y )  <->  ( x  =  A  ->  x  =  A ) ) )
54ralbidv 2843 . . . 4  |-  ( y  =  A  ->  ( A. x  e.  B  ( x  =  A  ->  x  =  y )  <->  A. x  e.  B  ( x  =  A  ->  x  =  A ) ) )
65spcegv 3145 . . 3  |-  ( A  e.  V  ->  ( A. x  e.  B  ( x  =  A  ->  x  =  A )  ->  E. y A. x  e.  B  ( x  =  A  ->  x  =  y ) ) )
72, 6mpi 20 . 2  |-  ( A  e.  V  ->  E. y A. x  e.  B  ( x  =  A  ->  x  =  y ) )
8 nfv 1728 . . 3  |-  F/ y  x  =  A
98rmo2 3366 . 2  |-  ( E* x  e.  B  x  =  A  <->  E. y A. x  e.  B  ( x  =  A  ->  x  =  y ) )
107, 9sylibr 212 1  |-  ( A  e.  V  ->  E* x  e.  B  x  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   E.wex 1633    e. wcel 1842   A.wral 2754   E*wrmo 2757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rmo 2762  df-v 3061
This theorem is referenced by: (None)
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