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Theorem euan 2200
Description: Introduction of a conjunct into uniqueness quantifier. (Contributed by NM, 19-Feb-2005.) (Proof shortened by Andrew Salmon, 9-Jul-2011.)
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 moanim.1 . . . . . 6  |-  F/ x ph
2 simpl 443 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
31, 2exlimi 1801 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ph )
43adantr 451 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ph )
5 simpr 447 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
65eximi 1563 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  E. x ps )
76adantr 451 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E. x ps )
8 nfe1 1706 . . . . . 6  |-  F/ x E. x ( ph  /\  ps )
93a1d 22 . . . . . . . 8  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ph ) )
109ancrd 537 . . . . . . 7  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ( ph  /\  ps ) ) )
115, 10impbid2 195 . . . . . 6  |-  ( E. x ( ph  /\  ps )  ->  ( (
ph  /\  ps )  <->  ps ) )
128, 11mobid 2177 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ( ph  /\  ps )  <->  E* x ps )
)
1312biimpa 470 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E* x ps )
144, 7, 13jca32 521 . . 3  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
15 eu5 2181 . . 3  |-  ( E! x ( ph  /\  ps )  <->  ( E. x
( ph  /\  ps )  /\  E* x ( ph  /\ 
ps ) ) )
16 eu5 2181 . . . 4  |-  ( E! x ps  <->  ( E. x ps  /\  E* x ps ) )
1716anbi2i 675 . . 3  |-  ( (
ph  /\  E! x ps )  <->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
1814, 15, 173imtr4i 257 . 2  |-  ( E! x ( ph  /\  ps )  ->  ( ph  /\  E! x ps )
)
19 ibar 490 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
201, 19eubid 2150 . . 3  |-  ( ph  ->  ( E! x ps  <->  E! x ( ph  /\  ps ) ) )
2120biimpa 470 . 2  |-  ( (
ph  /\  E! x ps )  ->  E! x
( ph  /\  ps )
)
2218, 21impbii 180 1  |-  ( E! x ( ph  /\  ps )  <->  ( ph  /\  E! x ps ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528   F/wnf 1531   E!weu 2143   E*wmo 2144
This theorem is referenced by:  euanv  2204  2eu7  2229  2eu8  2230
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-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148
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