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Theorem rmoxfrd 25882
Description: Transfer "at most one" restricted quantification from a variable  x to another variable  y contained in expression 
A. (Contributed by Thierry Arnoux, 7-Apr-2017.) (Revised by Thierry Arnoux, 8-Oct-2017.)
Hypotheses
Ref Expression
rmoxfrd.1  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
rmoxfrd.2  |-  ( (
ph  /\  x  e.  B )  ->  E! y  e.  C  x  =  A )
rmoxfrd.3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rmoxfrd  |-  ( ph  ->  ( E* x  e.  B  ps  <->  E* y  e.  C  ch )
)
Distinct variable groups:    x, A    x, y, B    x, C, y    ph, x, y    ps, y    ch, x
Allowed substitution hints:    ps( x)    ch( y)    A( y)

Proof of Theorem rmoxfrd
StepHypRef Expression
1 rmoxfrd.1 . . . . . 6  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
2 rmoxfrd.2 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  E! y  e.  C  x  =  A )
3 reurex 2942 . . . . . . 7  |-  ( E! y  e.  C  x  =  A  ->  E. y  e.  C  x  =  A )
42, 3syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
5 rmoxfrd.3 . . . . . 6  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
61, 4, 5rexxfrd 4512 . . . . 5  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. y  e.  C  ch )
)
7 df-rex 2726 . . . . 5  |-  ( E. x  e.  B  ps  <->  E. x ( x  e.  B  /\  ps )
)
8 df-rex 2726 . . . . 5  |-  ( E. y  e.  C  ch  <->  E. y ( y  e.  C  /\  ch )
)
96, 7, 83bitr3g 287 . . . 4  |-  ( ph  ->  ( E. x ( x  e.  B  /\  ps )  <->  E. y ( y  e.  C  /\  ch ) ) )
101, 2, 5reuxfr4d 25879 . . . . 5  |-  ( ph  ->  ( E! x  e.  B  ps  <->  E! y  e.  C  ch )
)
11 df-reu 2727 . . . . 5  |-  ( E! x  e.  B  ps  <->  E! x ( x  e.  B  /\  ps )
)
12 df-reu 2727 . . . . 5  |-  ( E! y  e.  C  ch  <->  E! y ( y  e.  C  /\  ch )
)
1310, 11, 123bitr3g 287 . . . 4  |-  ( ph  ->  ( E! x ( x  e.  B  /\  ps )  <->  E! y ( y  e.  C  /\  ch ) ) )
149, 13imbi12d 320 . . 3  |-  ( ph  ->  ( ( E. x
( x  e.  B  /\  ps )  ->  E! x ( x  e.  B  /\  ps )
)  <->  ( E. y
( y  e.  C  /\  ch )  ->  E! y ( y  e.  C  /\  ch )
) ) )
15 df-mo 2258 . . 3  |-  ( E* x ( x  e.  B  /\  ps )  <->  ( E. x ( x  e.  B  /\  ps )  ->  E! x ( x  e.  B  /\  ps ) ) )
16 df-mo 2258 . . 3  |-  ( E* y ( y  e.  C  /\  ch )  <->  ( E. y ( y  e.  C  /\  ch )  ->  E! y ( y  e.  C  /\  ch ) ) )
1714, 15, 163bitr4g 288 . 2  |-  ( ph  ->  ( E* x ( x  e.  B  /\  ps )  <->  E* y ( y  e.  C  /\  ch ) ) )
18 df-rmo 2728 . 2  |-  ( E* x  e.  B  ps  <->  E* x ( x  e.  B  /\  ps )
)
19 df-rmo 2728 . 2  |-  ( E* y  e.  C  ch  <->  E* y ( y  e.  C  /\  ch )
)
2017, 18, 193bitr4g 288 1  |-  ( ph  ->  ( E* x  e.  B  ps  <->  E* y  e.  C  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   E!weu 2253   E*wmo 2254   E.wrex 2721   E!wreu 2722   E*wrmo 2723
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-v 2979
This theorem is referenced by:  disjrdx  25938
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