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Theorem reuxfr4d 23930
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfrd 4707 (Contributed by Thierry Arnoux, 7-Apr-2017.)
Hypotheses
Ref Expression
reuxfr4d.1  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
reuxfr4d.2  |-  ( (
ph  /\  x  e.  B )  ->  E! y  e.  C  x  =  A )
reuxfr4d.3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
reuxfr4d  |-  ( ph  ->  ( E! x  e.  B  ps  <->  E! y  e.  C  ch )
)
Distinct variable groups:    x, y, ph    ps, y    ch, x    x, A    x, B, y   
x, C, y
Allowed substitution hints:    ps( x)    ch( y)    A( y)

Proof of Theorem reuxfr4d
StepHypRef Expression
1 reuxfr4d.2 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  E! y  e.  C  x  =  A )
2 reurex 2882 . . . . . 6  |-  ( E! y  e.  C  x  =  A  ->  E. y  e.  C  x  =  A )
31, 2syl 16 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
43biantrurd 495 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( ps 
<->  ( E. y  e.  C  x  =  A  /\  ps ) ) )
5 r19.41v 2821 . . . . . 6  |-  ( E. y  e.  C  ( x  =  A  /\  ps )  <->  ( E. y  e.  C  x  =  A  /\  ps ) )
6 reuxfr4d.3 . . . . . . . 8  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
76pm5.32da 623 . . . . . . 7  |-  ( ph  ->  ( ( x  =  A  /\  ps )  <->  ( x  =  A  /\  ch ) ) )
87rexbidv 2687 . . . . . 6  |-  ( ph  ->  ( E. y  e.  C  ( x  =  A  /\  ps )  <->  E. y  e.  C  ( x  =  A  /\  ch ) ) )
95, 8syl5bbr 251 . . . . 5  |-  ( ph  ->  ( ( E. y  e.  C  x  =  A  /\  ps )  <->  E. y  e.  C  ( x  =  A  /\  ch )
) )
109adantr 452 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( E. y  e.  C  x  =  A  /\  ps )  <->  E. y  e.  C  ( x  =  A  /\  ch )
) )
114, 10bitrd 245 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( ps 
<->  E. y  e.  C  ( x  =  A  /\  ch ) ) )
1211reubidva 2851 . 2  |-  ( ph  ->  ( E! x  e.  B  ps  <->  E! x  e.  B  E. y  e.  C  ( x  =  A  /\  ch )
) )
13 reuxfr4d.1 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
14 reurmo 2883 . . . 4  |-  ( E! y  e.  C  x  =  A  ->  E* y  e.  C x  =  A )
151, 14syl 16 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  C x  =  A )
1613, 15reuxfr3d 23929 . 2  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  C  ( x  =  A  /\  ch )  <->  E! y  e.  C  ch ) )
1712, 16bitrd 245 1  |-  ( ph  ->  ( E! x  e.  B  ps  <->  E! y  e.  C  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1649    e. wcel 1721   E.wrex 2667   E!wreu 2668   E*wrmo 2669
This theorem is referenced by:  rmoxfrdOLD  23932  rmoxfrd  23933
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-v 2918
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