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Theorem exsbOLD 2207
Description: An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
exsbOLD  |-  ( 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 exsbOLD
StepHypRef Expression
1 nfv 1629 . . 3  |-  F/ y
ph
21sb8e 2168 . 2  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
3 sb6 2174 . . 3  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
43exbii 1592 . 2  |-  ( E. y [ y  /  x ] ph  <->  E. y A. x ( x  =  y  ->  ph ) )
52, 4bitri 241 1  |-  ( E. x ph  <->  E. y A. x ( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550    = wceq 1652   [wsb 1658
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
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