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Theorem eupickb 2208
Description: Existential uniqueness "pick" showing wff equivalence. (Contributed by NM, 25-Nov-1994.)
Assertion
Ref Expression
eupickb  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph 
<->  ps ) )

Proof of Theorem eupickb
StepHypRef Expression
1 eupick 2206 . . 3  |-  ( ( E! x ph  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
213adant2 974 . 2  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph  ->  ps ) )
3 3simpc 954 . . 3  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( E! x ps  /\  E. x ( ph  /\  ps ) ) )
4 pm3.22 436 . . . . 5  |-  ( (
ph  /\  ps )  ->  ( ps  /\  ph ) )
54eximi 1563 . . . 4  |-  ( E. x ( ph  /\  ps )  ->  E. x
( ps  /\  ph ) )
65anim2i 552 . . 3  |-  ( ( E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( E! x ps  /\  E. x ( ps  /\  ph ) ) )
7 eupick 2206 . . 3  |-  ( ( E! x ps  /\  E. x ( ps  /\  ph ) )  ->  ( ps  ->  ph ) )
83, 6, 73syl 18 . 2  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ps  ->  ph ) )
92, 8impbid 183 1  |-  ( ( E! x ph  /\  E! x ps  /\  E. x ( ph  /\  ps ) )  ->  ( ph 
<->  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934   E.wex 1528   E!weu 2143
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-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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