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Theorem rmo2i 2905
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 2411 . 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
32rmo2ilem 2904 . 2  |-  ( E. y A. x  e.  A  ( ph  ->  x  =  y )  ->  E* x  e.  A  ph )
41, 3syl 14 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    = wceq 1285   F/wnf 1390   E.wex 1422   A.wral 2349   E.wrex 2350   E*wrmo 2352
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-ral 2354  df-rex 2355  df-rmo 2357
This theorem is referenced by: (None)
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