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Theorem ralxfrd 4661
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 714 . . . 4  |-  ( ( ( ph  /\  y  e.  C )  /\  x  =  A )  ->  ( ps 
<->  ch ) )
41, 3rspcdv 3217 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( A. x  e.  B  ps  ->  ch ) )
54ralrimdva 2882 . 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 2997 . . . . 5  |-  ( ( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  E. y  e.  C  ( ch  /\  x  =  A ) )
82biimprd 223 . . . . . . . . 9  |-  ( (
ph  /\  x  =  A )  ->  ( ch  ->  ps ) )
98expimpd 603 . . . . . . . 8  |-  ( ph  ->  ( ( x  =  A  /\  ch )  ->  ps ) )
109ancomsd 454 . . . . . . 7  |-  ( ph  ->  ( ( ch  /\  x  =  A )  ->  ps ) )
1110ad2antrr 725 . . . . . 6  |-  ( ( ( ph  /\  x  e.  B )  /\  y  e.  C )  ->  (
( ch  /\  x  =  A )  ->  ps ) )
1211rexlimdva 2955 . . . . 5  |-  ( (
ph  /\  x  e.  B )  ->  ( E. y  e.  C  ( ch  /\  x  =  A )  ->  ps ) )
137, 12syl5 32 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  (
( A. y  e.  C  ch  /\  E. y  e.  C  x  =  A )  ->  ps ) )
146, 13mpan2d 674 . . 3  |-  ( (
ph  /\  x  e.  B )  ->  ( A. y  e.  C  ch  ->  ps ) )
1514ralrimdva 2882 . 2  |-  ( ph  ->  ( A. y  e.  C  ch  ->  A. x  e.  B  ps )
)
165, 15impbid 191 1  |-  ( ph  ->  ( A. x  e.  B  ps  <->  A. y  e.  C  ch )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ral 2819  df-rex 2820  df-v 3115
This theorem is referenced by:  rexxfrd  4662  ralxfr2d  4663  ralxfr  4665  islindf4  18668  cmpfi  19702  metucnOLD  20854  metucn  20855  rlimcnp  23051  glbconN  34191  mapdordlem2  36452
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