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Theorem ralxfr2d 4566
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by Mario Carneiro, 20-Aug-2014.)
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
ralxfr2d  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. 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 ralxfr2d
StepHypRef Expression
1 ralxfr2d.1 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  V )
2 elisset 2811 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
31, 2syl 15 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  E. x  x  =  A )
4 ralxfr2d.2 . . . . . . . 8  |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A ) )
54biimprd 214 . . . . . . 7  |-  ( ph  ->  ( E. y  e.  C  x  =  A  ->  x  e.  B
) )
6 r19.23v 2672 . . . . . . 7  |-  ( A. y  e.  C  (
x  =  A  ->  x  e.  B )  <->  ( E. y  e.  C  x  =  A  ->  x  e.  B ) )
75, 6sylibr 203 . . . . . 6  |-  ( ph  ->  A. y  e.  C  ( x  =  A  ->  x  e.  B ) )
87r19.21bi 2654 . . . . 5  |-  ( (
ph  /\  y  e.  C )  ->  (
x  =  A  ->  x  e.  B )
)
9 eleq1 2356 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
108, 9mpbidi 207 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  (
x  =  A  ->  A  e.  B )
)
1110exlimdv 1626 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( E. x  x  =  A  ->  A  e.  B
) )
123, 11mpd 14 . 2  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
134biimpa 470 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
14 ralxfr2d.3 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1512, 13, 14ralxfrd 4564 1  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. y  e.  C  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557
This theorem is referenced by:  rexxfr2d  4567  ralrn  5684  ralima  5774  cnrest2  17030  cnprest2  17034  consuba  17162  subislly  17223  trfbas2  17554  trfil2  17598  flimrest  17694  fclsrest  17735  tsmssubm  17841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803
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