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Theorem reuxfrd 4559
Description: Transfer existential uniqueness from a variable  x to another variable  y contained in expression  A. Use reuhypd 4561 to eliminate the second hypothesis. (Contributed by NM, 16-Jan-2012.)
Hypotheses
Ref Expression
reuxfrd.1  |-  ( (
ph  /\  y  e.  B )  ->  A  e.  B )
reuxfrd.2  |-  ( (
ph  /\  x  e.  B )  ->  E! y  e.  B  x  =  A )
reuxfrd.3  |-  ( x  =  A  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
reuxfrd  |-  ( ph  ->  ( E! x  e.  B  ps  <->  E! y  e.  B  ch )
)
Distinct variable groups:    x, y, ph    ps, y    ch, x    x, A    x, B, y
Allowed substitution hints:    ps( x)    ch( y)    A( y)

Proof of Theorem reuxfrd
StepHypRef Expression
1 reuxfrd.2 . . . . . 6  |-  ( (
ph  /\  x  e.  B )  ->  E! y  e.  B  x  =  A )
2 reurex 2754 . . . . . 6  |-  ( E! y  e.  B  x  =  A  ->  E. y  e.  B  x  =  A )
31, 2syl 15 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  B  x  =  A )
43biantrurd 494 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  ( ps 
<->  ( E. y  e.  B  x  =  A  /\  ps ) ) )
5 r19.41v 2693 . . . . 5  |-  ( E. y  e.  B  ( x  =  A  /\  ps )  <->  ( E. y  e.  B  x  =  A  /\  ps ) )
6 reuxfrd.3 . . . . . . 7  |-  ( x  =  A  ->  ( ps 
<->  ch ) )
76pm5.32i 618 . . . . . 6  |-  ( ( x  =  A  /\  ps )  <->  ( x  =  A  /\  ch )
)
87rexbii 2568 . . . . 5  |-  ( E. y  e.  B  ( x  =  A  /\  ps )  <->  E. y  e.  B  ( x  =  A  /\  ch ) )
95, 8bitr3i 242 . . . 4  |-  ( ( E. y  e.  B  x  =  A  /\  ps )  <->  E. y  e.  B  ( x  =  A  /\  ch ) )
104, 9syl6bb 252 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( ps 
<->  E. y  e.  B  ( x  =  A  /\  ch ) ) )
1110reubidva 2723 . 2  |-  ( ph  ->  ( E! x  e.  B  ps  <->  E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ch )
) )
12 reuxfrd.1 . . 3  |-  ( (
ph  /\  y  e.  B )  ->  A  e.  B )
13 reurmo 2755 . . . 4  |-  ( E! y  e.  B  x  =  A  ->  E* y  e.  B x  =  A )
141, 13syl 15 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  E* y  e.  B x  =  A )
1512, 14reuxfr2d 4557 . 2  |-  ( ph  ->  ( E! x  e.  B  E. y  e.  B  ( x  =  A  /\  ch )  <->  E! y  e.  B  ch ) )
1611, 15bitrd 244 1  |-  ( ph  ->  ( E! x  e.  B  ps  <->  E! y  e.  B  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   E.wrex 2544   E!wreu 2545   E*wrmo 2546
This theorem is referenced by:  reuxfr  4560  riotaxfrd  6336
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-v 2790
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