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Theorem reuan 38473
Description: Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2329. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Hypothesis
Ref Expression
rmoanim.1  |-  F/ x ph
Assertion
Ref Expression
reuan  |-  ( E! x  e.  A  (
ph  /\  ps )  <->  (
ph  /\  E! x  e.  A  ps )
)
Distinct variable group:    x, A
Allowed substitution hints:    ph( x)    ps( x)

Proof of Theorem reuan
StepHypRef Expression
1 rmoanim.1 . . . . . 6  |-  F/ x ph
2 simpl 458 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ph )
32a1i 11 . . . . . 6  |-  ( x  e.  A  ->  (
( ph  /\  ps )  ->  ph ) )
41, 3rexlimi 2904 . . . . 5  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ph )
54adantr 466 . . . 4  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  E* x  e.  A  ( ph  /\  ps )
)  ->  ph )
6 simpr 462 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
76reximi 2890 . . . . 5  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ps )
87adantr 466 . . . 4  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  E* x  e.  A  ( ph  /\  ps )
)  ->  E. x  e.  A  ps )
9 nfre1 2883 . . . . . 6  |-  F/ x E. x  e.  A  ( ph  /\  ps )
104adantr 466 . . . . . . . . 9  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ph )
1110a1d 26 . . . . . . . 8  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ps  ->  ph ) )
1211ancrd 556 . . . . . . 7  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ps  ->  ( ph  /\  ps ) ) )
136, 12impbid2 207 . . . . . 6  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ( ph  /\  ps )  <->  ps )
)
149, 13rmobida 3013 . . . . 5  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E* x  e.  A  ( ph  /\  ps )  <->  E* x  e.  A  ps ) )
1514biimpa 486 . . . 4  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  E* x  e.  A  ( ph  /\  ps )
)  ->  E* x  e.  A  ps )
165, 8, 15jca32 537 . . 3  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  E* x  e.  A  ( ph  /\  ps )
)  ->  ( ph  /\  ( E. x  e.  A  ps  /\  E* x  e.  A  ps ) ) )
17 reu5 3043 . . 3  |-  ( E! x  e.  A  (
ph  /\  ps )  <->  ( E. x  e.  A  ( ph  /\  ps )  /\  E* x  e.  A  ( ph  /\  ps )
) )
18 reu5 3043 . . . 4  |-  ( E! x  e.  A  ps  <->  ( E. x  e.  A  ps  /\  E* x  e.  A  ps ) )
1918anbi2i 698 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  <->  (
ph  /\  ( E. x  e.  A  ps  /\ 
E* x  e.  A  ps ) ) )
2016, 17, 193imtr4i 269 . 2  |-  ( E! x  e.  A  (
ph  /\  ps )  ->  ( ph  /\  E! x  e.  A  ps ) )
21 ibar 506 . . . . 5  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
2221adantr 466 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ( ph  /\  ps ) ) )
231, 22reubida 3008 . . 3  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ( ph  /\ 
ps ) ) )
2423biimpa 486 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x  e.  A  ( ph  /\  ps )
)
2520, 24impbii 190 1  |-  ( E! x  e.  A  (
ph  /\  ps )  <->  (
ph  /\  E! x  e.  A  ps )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370   F/wnf 1661    e. wcel 1872   E.wrex 2772   E!wreu 2773   E*wrmo 2774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-12 1909
This theorem depends on definitions:  df-bi 188  df-an 372  df-ex 1658  df-nf 1662  df-eu 2273  df-mo 2274  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779
This theorem is referenced by:  2reu7  38484  2reu8  38485
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