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Theorem rexxfr2d 4425
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by Mario Carneiro, 20-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
ralxfr2d.1  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  V )
ralxfr2d.2  |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A ) )
ralxfr2d.3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
rexxfr2d  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. y  e.  C  ch )
)
Distinct variable groups:    x, A    x, y, B    x, C    ch, x    ph, x, y    ps, y
Allowed substitution hints:    ps( x)    ch( y)    A( y)    C( y)    V( x, y)

Proof of Theorem rexxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  V )
2 elisset 2726 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
31, 2syl 14 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  E. x  x  =  A )
4 ralxfr2d.2 . . . . . . . 8  |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A ) )
54biimprd 157 . . . . . . 7  |-  ( ph  ->  ( E. y  e.  C  x  =  A  ->  x  e.  B
) )
6 r19.23v 2566 . . . . . . 7  |-  ( A. y  e.  C  (
x  =  A  ->  x  e.  B )  <->  ( E. y  e.  C  x  =  A  ->  x  e.  B ) )
75, 6sylibr 133 . . . . . 6  |-  ( ph  ->  A. y  e.  C  ( x  =  A  ->  x  e.  B ) )
87r19.21bi 2545 . . . . 5  |-  ( (
ph  /\  y  e.  C )  ->  (
x  =  A  ->  x  e.  B )
)
9 eleq1 2220 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
108, 9mpbidi 150 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  (
x  =  A  ->  A  e.  B )
)
1110exlimdv 1799 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( E. x  x  =  A  ->  A  e.  B
) )
123, 11mpd 13 . 2  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
134biimpa 294 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
14 ralxfr2d.3 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1512, 13, 14rexxfrd 4423 1  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. y  e.  C  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1335   E.wex 1472    e. wcel 2128   A.wral 2435   E.wrex 2436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714
This theorem is referenced by:  rexrn  5604  rexima  5705  cnptopresti  12649  cnptoprest  12650  txrest  12687
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