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Theorem rmo2i 3343
Description: Condition implying restricted "at most one." (Contributed by NM, 17-Jun-2017.)
Hypothesis
Ref Expression
rmo2.1  |-  F/ y
ph
Assertion
Ref Expression
rmo2i  |-  ( E. y  e.  A  A. x  e.  A  ( ph  ->  x  =  y )  ->  E* x  e.  A  ph )
Distinct variable group:    x, y, A
Allowed substitution hints:    ph( x, y)

Proof of Theorem rmo2i
StepHypRef Expression
1 rexex 2843 . 2  |-  ( E. y  e.  A  A. x  e.  A  ( ph  ->  x  =  y )  ->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
2 rmo2.1 . . 3  |-  F/ y
ph
32rmo2 3342 . 2  |-  ( E* x  e.  A  ph  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
41, 3sylibr 217 1  |-  ( E. y  e.  A  A. x  e.  A  ( ph  ->  x  =  y )  ->  E* x  e.  A  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   E.wex 1671   F/wnf 1675   A.wral 2756   E.wrex 2757   E*wrmo 2759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-an 378  df-ex 1672  df-nf 1676  df-eu 2323  df-mo 2324  df-ral 2761  df-rex 2762  df-rmo 2764
This theorem is referenced by: (None)
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