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

Theorem reuxfr3d 27515
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Cf. reuxfr2d 4679 (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 3298 . . . . . . 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 3049 . . . . . 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 2871 . . . 4  |-  ( ph  ->  A. x  e.  B  E* y  e.  C  ( x  =  A  /\  ps ) )
8 2reuswap 3302 . . . 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 27514 . . . 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 3275 . . . . . . 7  |-  E* x  x  =  A
1211moani 2346 . . . . . 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 797 . . . . . . . 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 2308 . . . . . 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 2820 . . 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 3233 . . . 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 3041 . 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 1395    e. wcel 1819   E*wmo 2284   A.wral 2807   E.wrex 2808   E!wreu 2809   E*wrmo 2810
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-v 3111
This theorem is referenced by:  reuxfr4d  27516
  Copyright terms: Public domain W3C validator