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Theorem euan 1972
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
euan.1  |-  ( ph  ->  A. x ph )
Assertion
Ref Expression
euan  |-  ( E! x ( ph  /\  ps )  <->  ( ph  /\  E! x ps ) )

Proof of Theorem euan
StepHypRef Expression
1 euan.1 . . . . . 6  |-  ( ph  ->  A. x ph )
2 simpl 106 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
31, 2exlimih 1500 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ph )
43adantr 265 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ph )
5 simpr 107 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
65eximi 1507 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  E. x ps )
76adantr 265 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E. x ps )
8 hbe1 1400 . . . . . 6  |-  ( E. x ( ph  /\  ps )  ->  A. x E. x ( ph  /\  ps ) )
93a1d 22 . . . . . . . 8  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ph ) )
109ancrd 313 . . . . . . 7  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ( ph  /\  ps ) ) )
115, 10impbid2 135 . . . . . 6  |-  ( E. x ( ph  /\  ps )  ->  ( (
ph  /\  ps )  <->  ps ) )
128, 11mobidh 1950 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ( ph  /\  ps )  <->  E* x ps )
)
1312biimpa 284 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E* x ps )
144, 7, 13jca32 297 . . 3  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
15 eu5 1963 . . 3  |-  ( E! x ( ph  /\  ps )  <->  ( E. x
( ph  /\  ps )  /\  E* x ( ph  /\ 
ps ) ) )
16 eu5 1963 . . . 4  |-  ( E! x ps  <->  ( E. x ps  /\  E* x ps ) )
1716anbi2i 438 . . 3  |-  ( (
ph  /\  E! x ps )  <->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
1814, 15, 173imtr4i 194 . 2  |-  ( E! x ( ph  /\  ps )  ->  ( ph  /\  E! x ps )
)
19 ibar 289 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
201, 19eubidh 1922 . . 3  |-  ( ph  ->  ( E! x ps  <->  E! x ( ph  /\  ps ) ) )
2120biimpa 284 . 2  |-  ( (
ph  /\  E! x ps )  ->  E! x
( ph  /\  ps )
)
2218, 21impbii 121 1  |-  ( E! x ( ph  /\  ps )  <->  ( ph  /\  E! x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102   A.wal 1257   E.wex 1397   E!weu 1916   E*wmo 1917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920
This theorem is referenced by:  euanv  1973  2eu7  2010
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