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Theorem euan 2004
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 107 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ph )
31, 2exlimih 1529 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ph )
43adantr 270 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ph )
5 simpr 108 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
65eximi 1536 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  E. x ps )
76adantr 270 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E. x ps )
8 hbe1 1429 . . . . . 6  |-  ( E. x ( ph  /\  ps )  ->  A. x E. x ( ph  /\  ps ) )
93a1d 22 . . . . . . . 8  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ph ) )
109ancrd 319 . . . . . . 7  |-  ( E. x ( ph  /\  ps )  ->  ( ps 
->  ( ph  /\  ps ) ) )
115, 10impbid2 141 . . . . . 6  |-  ( E. x ( ph  /\  ps )  ->  ( (
ph  /\  ps )  <->  ps ) )
128, 11mobidh 1982 . . . . 5  |-  ( E. x ( ph  /\  ps )  ->  ( E* x ( ph  /\  ps )  <->  E* x ps )
)
1312biimpa 290 . . . 4  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  E* x ps )
144, 7, 13jca32 303 . . 3  |-  ( ( E. x ( ph  /\ 
ps )  /\  E* x ( ph  /\  ps ) )  ->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
15 eu5 1995 . . 3  |-  ( E! x ( ph  /\  ps )  <->  ( E. x
( ph  /\  ps )  /\  E* x ( ph  /\ 
ps ) ) )
16 eu5 1995 . . . 4  |-  ( E! x ps  <->  ( E. x ps  /\  E* x ps ) )
1716anbi2i 445 . . 3  |-  ( (
ph  /\  E! x ps )  <->  ( ph  /\  ( E. x ps  /\  E* x ps ) ) )
1814, 15, 173imtr4i 199 . 2  |-  ( E! x ( ph  /\  ps )  ->  ( ph  /\  E! x ps )
)
19 ibar 295 . . . 4  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
201, 19eubidh 1954 . . 3  |-  ( ph  ->  ( E! x ps  <->  E! x ( ph  /\  ps ) ) )
2120biimpa 290 . 2  |-  ( (
ph  /\  E! x ps )  ->  E! x
( ph  /\  ps )
)
2218, 21impbii 124 1  |-  ( E! x ( ph  /\  ps )  <->  ( ph  /\  E! x ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    <-> wb 103   A.wal 1287   E.wex 1426   E!weu 1948   E*wmo 1949
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952
This theorem is referenced by:  euanv  2005  2eu7  2042
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