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Theorem ralxfrd 4737
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by NM, 15-Aug-2014.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Hypotheses
Ref Expression
ralxfrd.1  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
ralxfrd.2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
ralxfrd.3  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
Assertion
Ref Expression
ralxfrd  |-  ( 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)

Proof of Theorem ralxfrd
StepHypRef Expression
1 ralxfrd.1 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
2 ralxfrd.3 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
32adantlr 696 . . . 4  |-  ( ( ( ph  /\  y  e.  C )  /\  x  =  A )  ->  ( ps 
<->  ch ) )
41, 3rspcdv 3055 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( A. x  e.  B  ps  ->  ch ) )
54ralrimdva 2796 . 2  |-  ( ph  ->  ( A. x  e.  B  ps  ->  A. y  e.  C  ch )
)
6 ralxfrd.2 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
7 r19.29 2846 . . . . 5  |-  ( ( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  E. y  e.  C  ( ch  /\  x  =  A ) )
82biimprd 215 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
98expimpd 587 . . . . . . . 8  |-  ( ph  ->  ( ( x  =  A  /\  ch )  ->  ps ) )
109ancomsd 441 . . . . . . 7  |-  ( ph  ->  ( ( ch  /\  x  =  A )  ->  ps ) )
1110ad2antrr 707 . . . . . 6  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  C )  ->  (
( ch  /\  x  =  A )  ->  ps ) )
1211rexlimdva 2830 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( E. y  e.  C  ( ch  /\  x  =  A )  ->  ps ) )
137, 12syl5 30 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  ps ) )
146, 13mpan2d 656 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( A. y  e.  C  ch  ->  ps ) )
1514ralrimdva 2796 . 2  |-  ( ph  ->  ( A. y  e.  C  ch  ->  A. x  e.  B  ps )
)
165, 15impbid 184 1  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. y  e.  C  ch )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   A.wral 2705   E.wrex 2706
This theorem is referenced by:  rexxfrd  4738  ralxfr2d  4739  ralxfr  4741  cmpfi  17471  metucnOLD  18618  metucn  18619  rlimcnp  20804  islindf4  27285  glbconN  30174  mapdordlem2  32435
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 2417
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-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958
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