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Theorem euanOLD 2339
Description: Obsolete poof of euan 2338 as of 24-Dec-2018. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.) (New usage is discouraged.) (Proof modification is discouraged.)
Hypothesis
Ref Expression
moanim.1  |-  F/ x ph
Assertion
Ref Expression
euanOLD  |-  ( E! x ( ph  /\  ps )  <->  ( ph  /\  E! x ps ) )

Proof of Theorem euanOLD
StepHypRef Expression
1 moanim.1 . . . . . 6  |-  F/ x ph
2 simpl 457 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
31, 2exlimi 1847 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ph )
43adantr 465 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ph )
5 exsimpr 1646 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  E. x ps )
65adantr 465 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E. x ps )
7 nfe1 1780 . . . . . 6  |-  F/ x E. x ( ph  /\  ps )
8 simpr 461 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ps )
93a1d 25 . . . . . . . 8  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ph ) )
109ancrd 554 . . . . . . 7  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ( ph  /\  ps ) ) )
118, 10impbid2 204 . . . . . 6  |-  ( E. x ( ph  /\  ps )  ->  ( (
ph  /\  ps )  <->  ps ) )
127, 11mobid 2282 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ( ph  /\  ps )  <->  E* x ps )
)
1312biimpa 484 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E* x ps )
144, 6, 13jca32 535 . . 3  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
15 eu5 2290 . . 3  |-  ( E! x ( ph  /\  ps )  <->  ( E. x
( ph  /\  ps )  /\  E* x ( ph  /\ 
ps ) ) )
16 eu5 2290 . . . 4  |-  ( E! x ps  <->  ( E. x ps  /\  E* x ps ) )
1716anbi2i 694 . . 3  |-  ( (
ph  /\  E! x ps )  <->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
1814, 15, 173imtr4i 266 . 2  |-  ( E! x ( ph  /\  ps )  ->  ( ph  /\  E! x ps )
)
19 ibar 504 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
201, 19eubid 2281 . . 3  |-  ( ph  ->  ( E! x ps  <->  E! x ( ph  /\  ps ) ) )
2120biimpa 484 . 2  |-  ( (
ph  /\  E! x ps )  ->  E! x
( ph  /\  ps )
)
2218, 21impbii 188 1  |-  ( E! x ( ph  /\  ps )  <->  ( ph  /\  E! x ps ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369   E.wex 1587   F/wnf 1590   E!weu 2260   E*wmo 2261
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-12 1794
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1588  df-nf 1591  df-eu 2264  df-mo 2265
This theorem is referenced by: (None)
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