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Theorem euexALT 2310
Description: Alternate proof of euex 2293. Shorter but uses more axioms. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
euexALT  |-  ( E! x ph  ->  E. x ph )

Proof of Theorem euexALT
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 nfv 1755 . . 3  |-  F/ y
ph
21eu1 2309 . 2  |-  ( E! x ph  <->  E. x
( ph  /\  A. y
( [ y  /  x ] ph  ->  x  =  y ) ) )
3 exsimpl 1723 . 2  |-  ( E. x ( ph  /\  A. y ( [ y  /  x ] ph  ->  x  =  y ) )  ->  E. x ph )
42, 3sylbi 198 1  |-  ( E! x ph  ->  E. x ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370   A.wal 1435   E.wex 1657   [wsb 1790   E!weu 2269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273
This theorem is referenced by: (None)
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