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Theorem exsb 1926
Description: An equivalent expression for existence. (Contributed by NM, 2-Feb-2005.)
Assertion
Ref Expression
exsb  |-  ( E. x ph  <->  E. y A. x ( x  =  y  ->  ph ) )
Distinct variable groups:    x, y    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem exsb
StepHypRef Expression
1 ax-17 1460 . . 3  |-  ( ph  ->  A. y ph )
21sb8eh 1777 . 2  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
3 sb6 1808 . . 3  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
43exbii 1537 . 2  |-  ( E. y [ y  /  x ] ph  <->  E. y A. x ( x  =  y  ->  ph ) )
52, 4bitri 182 1  |-  ( E. x ph  <->  E. y A. x ( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103   A.wal 1283   E.wex 1422   [wsb 1686
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-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-11 1438  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-sb 1687
This theorem is referenced by:  2exsb  1927
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