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Theorem reuan 27958
Description: Introduction of a conjunct into restricted uniqueness quantifier, analogous to euan 2200. (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 443 . . . . . . 7  |-  ( (
ph  /\  ps )  ->  ph )
32a1i 10 . . . . . 6  |-  ( x  e.  A  ->  (
( ph  /\  ps )  ->  ph ) )
41, 3rexlimi 2660 . . . . 5  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ph )
54adantr 451 . . . 4  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  E* x  e.  A
( ph  /\  ps )
)  ->  ph )
6 simpr 447 . . . . . 6  |-  ( (
ph  /\  ps )  ->  ps )
76reximi 2650 . . . . 5  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  E. x  e.  A  ps )
87adantr 451 . . . 4  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  E* x  e.  A
( ph  /\  ps )
)  ->  E. x  e.  A  ps )
9 nfre1 2599 . . . . . 6  |-  F/ x E. x  e.  A  ( ph  /\  ps )
104adantr 451 . . . . . . . . 9  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ph )
1110a1d 22 . . . . . . . 8  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ps  ->  ph ) )
1211ancrd 537 . . . . . . 7  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ps  ->  ( ph  /\  ps ) ) )
136, 12impbid2 195 . . . . . 6  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  x  e.  A
)  ->  ( ( ph  /\  ps )  <->  ps )
)
149, 13rmobida 2727 . . . . 5  |-  ( E. x  e.  A  (
ph  /\  ps )  ->  ( E* x  e.  A ( ph  /\  ps )  <->  E* x  e.  A ps ) )
1514biimpa 470 . . . 4  |-  ( ( E. x  e.  A  ( ph  /\  ps )  /\  E* x  e.  A
( ph  /\  ps )
)  ->  E* x  e.  A ps )
165, 8, 15jca32 521 . . 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 2753 . . 3  |-  ( E! x  e.  A  (
ph  /\  ps )  <->  ( E. x  e.  A  ( ph  /\  ps )  /\  E* x  e.  A
( ph  /\  ps )
) )
18 reu5 2753 . . . 4  |-  ( E! x  e.  A  ps  <->  ( E. x  e.  A  ps  /\  E* x  e.  A ps ) )
1918anbi2i 675 . . 3  |-  ( (
ph  /\  E! x  e.  A  ps )  <->  (
ph  /\  ( E. x  e.  A  ps  /\ 
E* x  e.  A ps ) ) )
2016, 17, 193imtr4i 257 . 2  |-  ( E! x  e.  A  (
ph  /\  ps )  ->  ( ph  /\  E! x  e.  A  ps ) )
21 ibar 490 . . . . 5  |-  ( ph  ->  ( ps  <->  ( ph  /\ 
ps ) ) )
2221adantr 451 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  ( ps 
<->  ( ph  /\  ps ) ) )
231, 22reubida 2722 . . 3  |-  ( ph  ->  ( E! x  e.  A  ps  <->  E! x  e.  A  ( ph  /\ 
ps ) ) )
2423biimpa 470 . 2  |-  ( (
ph  /\  E! x  e.  A  ps )  ->  E! x  e.  A  ( ph  /\  ps )
)
2520, 24impbii 180 1  |-  ( E! x  e.  A  (
ph  /\  ps )  <->  (
ph  /\  E! x  e.  A  ps )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   F/wnf 1531    e. wcel 1684   E.wrex 2544   E!wreu 2545   E*wrmo 2546
This theorem is referenced by:  2reu7  27969  2reu8  27970
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  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551
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