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Theorem exsbOLD 2070
Description: An equivalent expression for existence. Obsolete as of 19-Jun-2017. (Contributed by NM, 2-Feb-2005.) (New usage 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 1605 . . 3  |-  F/ y
ph
21sb8e 2033 . 2  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
3 sb6 2038 . . 3  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
43exbii 1569 . 2  |-  ( E. y [ y  /  x ] ph  <->  E. y A. x ( x  =  y  ->  ph ) )
52, 4bitri 240 1  |-  ( E. x ph  <->  E. y A. x ( x  =  y  ->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1527   E.wex 1528    = wceq 1623   [wsb 1629
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
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