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Theorem reuxfr3d 27960
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfr2d 4645 (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
reuxfr3d.1  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
reuxfr3d.2  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  C  x  =  A )
Assertion
Ref Expression
reuxfr3d  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  C  ( x  =  A  /\  ps )  <->  E! y  e.  C  ps ) )
Distinct variable groups:    x, y, ph    ps, x    x, A    x, B, y    x, C, y
Allowed substitution hints:    ps( y)    A( y)

Proof of Theorem reuxfr3d
StepHypRef Expression
1 reuxfr3d.2 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  C  x  =  A )
2 rmoan 3276 . . . . . . 7  |-  ( E* y  e.  C  x  =  A  ->  E* y  e.  C  ( ps  /\  x  =  A ) )
31, 2syl 17 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  C  ( ps  /\  x  =  A ) )
4 ancom 451 . . . . . . 7  |-  ( ( ps  /\  x  =  A )  <->  ( x  =  A  /\  ps )
)
54rmobii 3027 . . . . . 6  |-  ( E* y  e.  C  ( ps  /\  x  =  A )  <->  E* y  e.  C  ( x  =  A  /\  ps )
)
63, 5sylib 199 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  C  (
x  =  A  /\  ps ) )
76ralrimiva 2846 . . . 4  |-  ( ph  ->  A. x  e.  B  E* y  e.  C  ( x  =  A  /\  ps ) )
8 2reuswap 3280 . . . 4  |-  ( A. x  e.  B  E* y  e.  C  (
x  =  A  /\  ps )  ->  ( E! x  e.  B  E. y  e.  C  (
x  =  A  /\  ps )  ->  E! y  e.  C  E. x  e.  B  ( x  =  A  /\  ps )
) )
97, 8syl 17 . . 3  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  C  ( x  =  A  /\  ps )  ->  E! y  e.  C  E. x  e.  B  ( x  =  A  /\  ps ) ) )
10 2reuswap2 27959 . . . 4  |-  ( A. y  e.  C  E* x ( x  e.  B  /\  ( x  =  A  /\  ps ) )  ->  ( E! y  e.  C  E. x  e.  B  ( x  =  A  /\  ps )  ->  E! x  e.  B  E. y  e.  C  (
x  =  A  /\  ps ) ) )
11 moeq 3253 . . . . . . 7  |-  E* x  x  =  A
1211moani 2324 . . . . . 6  |-  E* x
( ( x  e.  B  /\  ps )  /\  x  =  A
)
13 ancom 451 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  ps )  /\  x  =  A )  <->  ( x  =  A  /\  (
x  e.  B  /\  ps ) ) )
14 an12 804 . . . . . . . 8  |-  ( ( x  =  A  /\  ( x  e.  B  /\  ps ) )  <->  ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
1513, 14bitri 252 . . . . . . 7  |-  ( ( ( x  e.  B  /\  ps )  /\  x  =  A )  <->  ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
1615mobii 2291 . . . . . 6  |-  ( E* x ( ( x  e.  B  /\  ps )  /\  x  =  A )  <->  E* x ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
1712, 16mpbi 211 . . . . 5  |-  E* x
( x  e.  B  /\  ( x  =  A  /\  ps ) )
1817a1i 11 . . . 4  |-  ( y  e.  C  ->  E* x ( x  e.  B  /\  ( x  =  A  /\  ps ) ) )
1910, 18mprg 2795 . . 3  |-  ( E! y  e.  C  E. x  e.  B  (
x  =  A  /\  ps )  ->  E! x  e.  B  E. y  e.  C  ( x  =  A  /\  ps )
)
209, 19impbid1 206 . 2  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  C  ( x  =  A  /\  ps )  <->  E! y  e.  C  E. x  e.  B  (
x  =  A  /\  ps ) ) )
21 reuxfr3d.1 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
22 biidd 240 . . . . 5  |-  ( x  =  A  ->  ( ps 
<->  ps ) )
2322ceqsrexv 3211 . . . 4  |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ps )  <->  ps )
)
2421, 23syl 17 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( E. x  e.  B  ( x  =  A  /\  ps )  <->  ps )
)
2524reubidva 3019 . 2  |-  ( ph  ->  ( E! y  e.  C  E. x  e.  B  ( x  =  A  /\  ps )  <->  E! y  e.  C  ps ) )
2620, 25bitrd 256 1  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  C  ( x  =  A  /\  ps )  <->  E! y  e.  C  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870   E*wmo 2267   A.wral 2782   E.wrex 2783   E!wreu 2784   E*wrmo 2785
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-v 3089
This theorem is referenced by:  reuxfr4d  27961
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