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Theorem euan 2358
Description: Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (Proof shortened by Wolf Lammen, 24-Dec-2018.)
Hypothesis
Ref Expression
moanim.1  |-  F/ x ph
Assertion
Ref Expression
euan  |-  ( E! x ( ph  /\  ps )  <->  ( ph  /\  E! x ps ) )

Proof of Theorem euan
StepHypRef Expression
1 euex 2322 . . . 4  |-  ( E! x ( ph  /\  ps )  ->  E. x
( ph  /\  ps )
)
2 moanim.1 . . . . 5  |-  F/ x ph
3 simpl 459 . . . . 5  |-  ( (
ph  /\  ps )  ->  ph )
42, 3exlimi 1994 . . . 4  |-  ( E. x ( ph  /\  ps )  ->  ph )
51, 4syl 17 . . 3  |-  ( E! x ( ph  /\  ps )  ->  ph )
6 ibar 507 . . . . 5  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
72, 6eubid 2316 . . . 4  |-  ( ph  ->  ( E! x ps  <->  E! x ( ph  /\  ps ) ) )
87biimprcd 229 . . 3  |-  ( E! x ( ph  /\  ps )  ->  ( ph  ->  E! x ps )
)
95, 8jcai 539 . 2  |-  ( E! x ( ph  /\  ps )  ->  ( ph  /\  E! x ps )
)
107biimpa 487 . 2  |-  ( (
ph  /\  E! x ps )  ->  E! x
( ph  /\  ps )
)
119, 10impbii 191 1  |-  ( E! x ( ph  /\  ps )  <->  ( ph  /\  E! x ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    /\ wa 371   E.wex 1662   F/wnf 1666   E!weu 2298
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-10 1914  ax-12 1932
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1663  df-nf 1667  df-eu 2302
This theorem is referenced by:  euanv  2362  2eu7  2387  2eu8  2388
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