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Theorem rmo2i 3090
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 2615 . 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 3089 . 2  |-  ( E* x  e.  A ph  <->  E. y A. x  e.  A  ( ph  ->  x  =  y ) )
41, 3sylibr 203 1  |-  ( E. y  e.  A  A. x  e.  A  ( ph  ->  x  =  y )  ->  E* x  e.  A ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1531   F/wnf 1534   A.wral 2556   E.wrex 2557   E*wrmo 2559
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-ral 2561  df-rex 2562  df-rmo 2564
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