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Theorem rmo2i 2971
Description: Condition implying restricted at-most-one quantifier. (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 2456 . 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 2970 . 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 1316   F/wnf 1421   E.wex 1453   A.wral 2393   E.wrex 2394   E*wrmo 2396
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500
This theorem depends on definitions:  df-bi 116  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-ral 2398  df-rex 2399  df-rmo 2401
This theorem is referenced by: (None)
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