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Theorem exsbOLD 2083
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 1609 . . 3  |-  F/ y
ph
21sb8e 2046 . 2  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
3 sb6 2051 . . 3  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
43exbii 1572 . 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 1530   E.wex 1531    = wceq 1632   [wsb 1638
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639
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