Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  reuxfr3d Structured version   Unicode version

Theorem reuxfr3d 26020
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfr2d 4618 (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 3259 . . . . . . 7  |-  ( E* y  e.  C  x  =  A  ->  E* y  e.  C  ( ps  /\  x  =  A ) )
31, 2syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  C  ( ps  /\  x  =  A ) )
4 ancom 450 . . . . . . 7  |-  ( ( ps  /\  x  =  A )  <->  ( x  =  A  /\  ps )
)
54rmobii 3012 . . . . . 6  |-  ( E* y  e.  C  ( ps  /\  x  =  A )  <->  E* y  e.  C  ( x  =  A  /\  ps )
)
63, 5sylib 196 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  C  (
x  =  A  /\  ps ) )
76ralrimiva 2827 . . . 4  |-  ( ph  ->  A. x  e.  B  E* y  e.  C  ( x  =  A  /\  ps ) )
8 2reuswap 3263 . . . 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 16 . . 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 26019 . . . 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 3236 . . . . . . 7  |-  E* x  x  =  A
1211moani 2334 . . . . . 6  |-  E* x
( ( x  e.  B  /\  ps )  /\  x  =  A
)
13 ancom 450 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  ps )  /\  x  =  A )  <->  ( x  =  A  /\  (
x  e.  B  /\  ps ) ) )
14 an12 795 . . . . . . . 8  |-  ( ( x  =  A  /\  ( x  e.  B  /\  ps ) )  <->  ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
1513, 14bitri 249 . . . . . . 7  |-  ( ( ( x  e.  B  /\  ps )  /\  x  =  A )  <->  ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
1615mobii 2287 . . . . . 6  |-  ( E* x ( ( x  e.  B  /\  ps )  /\  x  =  A )  <->  E* x ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
1712, 16mpbi 208 . . . . 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 2897 . . 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 203 . 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 237 . . . . 5  |-  ( x  =  A  ->  ( ps 
<->  ps ) )
2322ceqsrexv 3194 . . . 4  |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ps )  <->  ps )
)
2421, 23syl 16 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( E. x  e.  B  ( x  =  A  /\  ps )  <->  ps )
)
2524reubidva 3004 . 2  |-  ( ph  ->  ( E! y  e.  C  E. x  e.  B  ( x  =  A  /\  ps )  <->  E! y  e.  C  ps ) )
2620, 25bitrd 253 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 184    /\ wa 369    = wceq 1370    e. wcel 1758   E*wmo 2262   A.wral 2796   E.wrex 2797   E!wreu 2798   E*wrmo 2799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-ral 2801  df-rex 2802  df-reu 2803  df-rmo 2804  df-v 3074
This theorem is referenced by:  reuxfr4d  26021
  Copyright terms: Public domain W3C validator