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Theorem reuxfr2d 4738
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 16-Jan-2012.) (Revised by NM, 16-Jun-2017.)
Hypotheses
Ref Expression
reuxfr2d.1  |-  ( (
ph  /\  y  e.  B )  ->  A  e.  B )
reuxfr2d.2  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  B x  =  A )
Assertion
Ref Expression
reuxfr2d  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ps )  <->  E! y  e.  B  ps ) )
Distinct variable groups:    x, y, ph    ps, x    x, A    x, B, y
Allowed substitution hints:    ps( y)    A( y)

Proof of Theorem reuxfr2d
StepHypRef Expression
1 reuxfr2d.2 . . . . . . 7  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  B x  =  A )
2 rmoan 3124 . . . . . . 7  |-  ( E* y  e.  B x  =  A  ->  E* y  e.  B ( ps  /\  x  =  A ) )
31, 2syl 16 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  B ( ps  /\  x  =  A ) )
4 ancom 438 . . . . . . 7  |-  ( ( ps  /\  x  =  A )  <->  ( x  =  A  /\  ps )
)
54rmobii 2891 . . . . . 6  |-  ( E* y  e.  B ( ps  /\  x  =  A )  <->  E* y  e.  B ( x  =  A  /\  ps )
)
63, 5sylib 189 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  B (
x  =  A  /\  ps ) )
76ralrimiva 2781 . . . 4  |-  ( ph  ->  A. x  e.  B  E* y  e.  B
( x  =  A  /\  ps ) )
8 2reuswap 3128 . . . 4  |-  ( A. x  e.  B  E* y  e.  B (
x  =  A  /\  ps )  ->  ( E! x  e.  B  E. y  e.  B  (
x  =  A  /\  ps )  ->  E! y  e.  B  E. x  e.  B  ( x  =  A  /\  ps )
) )
97, 8syl 16 . . 3  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ps )  ->  E! y  e.  B  E. x  e.  B  ( x  =  A  /\  ps ) ) )
10 df-rmo 2705 . . . . . 6  |-  ( E* x  e.  B ( x  =  A  /\  ps )  <->  E* x ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
1110ralbii 2721 . . . . 5  |-  ( A. y  e.  B  E* x  e.  B (
x  =  A  /\  ps )  <->  A. y  e.  B  E* x ( x  e.  B  /\  ( x  =  A  /\  ps ) ) )
12 2reuswap 3128 . . . . 5  |-  ( A. y  e.  B  E* x  e.  B (
x  =  A  /\  ps )  ->  ( E! y  e.  B  E. x  e.  B  (
x  =  A  /\  ps )  ->  E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ps )
) )
1311, 12sylbir 205 . . . 4  |-  ( A. y  e.  B  E* x ( x  e.  B  /\  ( x  =  A  /\  ps ) )  ->  ( E! y  e.  B  E. x  e.  B  ( x  =  A  /\  ps )  ->  E! x  e.  B  E. y  e.  B  (
x  =  A  /\  ps ) ) )
14 moeq 3102 . . . . . . 7  |-  E* x  x  =  A
1514moani 2332 . . . . . 6  |-  E* x
( ( x  e.  B  /\  ps )  /\  x  =  A
)
16 ancom 438 . . . . . . . 8  |-  ( ( ( x  e.  B  /\  ps )  /\  x  =  A )  <->  ( x  =  A  /\  (
x  e.  B  /\  ps ) ) )
17 an12 773 . . . . . . . 8  |-  ( ( x  =  A  /\  ( x  e.  B  /\  ps ) )  <->  ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
1816, 17bitri 241 . . . . . . 7  |-  ( ( ( x  e.  B  /\  ps )  /\  x  =  A )  <->  ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
1918mobii 2316 . . . . . 6  |-  ( E* x ( ( x  e.  B  /\  ps )  /\  x  =  A )  <->  E* x ( x  e.  B  /\  (
x  =  A  /\  ps ) ) )
2015, 19mpbi 200 . . . . 5  |-  E* x
( x  e.  B  /\  ( x  =  A  /\  ps ) )
2120a1i 11 . . . 4  |-  ( y  e.  B  ->  E* x ( x  e.  B  /\  ( x  =  A  /\  ps ) ) )
2213, 21mprg 2767 . . 3  |-  ( E! y  e.  B  E. x  e.  B  (
x  =  A  /\  ps )  ->  E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ps )
)
239, 22impbid1 195 . 2  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ps )  <->  E! y  e.  B  E. x  e.  B  (
x  =  A  /\  ps ) ) )
24 reuxfr2d.1 . . . 4  |-  ( (
ph  /\  y  e.  B )  ->  A  e.  B )
25 biidd 229 . . . . 5  |-  ( x  =  A  ->  ( ps 
<->  ps ) )
2625ceqsrexv 3061 . . . 4  |-  ( A  e.  B  ->  ( E. x  e.  B  ( x  =  A  /\  ps )  <->  ps )
)
2724, 26syl 16 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  ( E. x  e.  B  ( x  =  A  /\  ps )  <->  ps )
)
2827reubidva 2883 . 2  |-  ( ph  ->  ( E! y  e.  B  E. x  e.  B  ( x  =  A  /\  ps )  <->  E! y  e.  B  ps ) )
2923, 28bitrd 245 1  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ps )  <->  E! y  e.  B  ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E*wmo 2281   A.wral 2697   E.wrex 2698   E!wreu 2699   E*wrmo 2700
This theorem is referenced by:  reuxfr2  4739  reuxfrd  4740
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-v 2950
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