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Theorem exsbOLD 2165
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 1626 . . 3  |-  F/ y
ph
21sb8e 2126 . 2  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
3 sb6 2132 . . 3  |-  ( [ y  /  x ] ph 
<-> 
A. x ( x  =  y  ->  ph )
)
43exbii 1589 . 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 1546   E.wex 1547    = wceq 1649   [wsb 1655
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
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