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Theorem euan 2031
Description: Introduction of a conjunct into unique existential 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 108 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
31, 2exlimih 1555 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ph )
43adantr 272 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ph )
5 simpr 109 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
65eximi 1562 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  E. x ps )
76adantr 272 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E. x ps )
8 hbe1 1454 . . . . . 6  |-  ( E. x ( ph  /\  ps )  ->  A. x E. x ( ph  /\  ps ) )
93a1d 22 . . . . . . . 8  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ph ) )
109ancrd 322 . . . . . . 7  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ( ph  /\  ps ) ) )
115, 10impbid2 142 . . . . . 6  |-  ( E. x ( ph  /\  ps )  ->  ( (
ph  /\  ps )  <->  ps ) )
128, 11mobidh 2009 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ( ph  /\  ps )  <->  E* x ps )
)
1312biimpa 292 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E* x ps )
144, 7, 13jca32 306 . . 3  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
15 eu5 2022 . . 3  |-  ( E! x ( ph  /\  ps )  <->  ( E. x
( ph  /\  ps )  /\  E* x ( ph  /\ 
ps ) ) )
16 eu5 2022 . . . 4  |-  ( E! x ps  <->  ( E. x ps  /\  E* x ps ) )
1716anbi2i 450 . . 3  |-  ( (
ph  /\  E! x ps )  <->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
1814, 15, 173imtr4i 200 . 2  |-  ( E! x ( ph  /\  ps )  ->  ( ph  /\  E! x ps )
)
19 ibar 297 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
201, 19eubidh 1981 . . 3  |-  ( ph  ->  ( E! x ps  <->  E! x ( ph  /\  ps ) ) )
2120biimpa 292 . 2  |-  ( (
ph  /\  E! x ps )  ->  E! x
( ph  /\  ps )
)
2218, 21impbii 125 1  |-  ( E! x ( ph  /\  ps )  <->  ( ph  /\  E! x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104   A.wal 1312   E.wex 1451   E!weu 1975   E*wmo 1976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498
This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979
This theorem is referenced by:  euanv  2032  2eu7  2069
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