MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ralxfr2d Structured version   Unicode version

Theorem ralxfr2d 4731
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 2958 . . . 4  |-  ( A  e.  V  ->  E. x  x  =  A )
31, 2syl 16 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  E. x  x  =  A )
4 ralxfr2d.2 . . . . . . . 8  |-  ( ph  ->  ( x  e.  B  <->  E. y  e.  C  x  =  A ) )
54biimprd 215 . . . . . . 7  |-  ( ph  ->  ( E. y  e.  C  x  =  A  ->  x  e.  B
) )
6 r19.23v 2814 . . . . . . 7  |-  ( A. y  e.  C  (
x  =  A  ->  x  e.  B )  <->  ( E. y  e.  C  x  =  A  ->  x  e.  B ) )
75, 6sylibr 204 . . . . . 6  |-  ( ph  ->  A. y  e.  C  ( x  =  A  ->  x  e.  B ) )
87r19.21bi 2796 . . . . 5  |-  ( (
ph  /\  y  e.  C )  ->  (
x  =  A  ->  x  e.  B )
)
9 eleq1 2495 . . . . 5  |-  ( x  =  A  ->  (
x  e.  B  <->  A  e.  B ) )
108, 9mpbidi 208 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  (
x  =  A  ->  A  e.  B )
)
1110exlimdv 1646 . . 3  |-  ( (
ph  /\  y  e.  C )  ->  ( E. x  x  =  A  ->  A  e.  B
) )
123, 11mpd 15 . 2  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
134biimpa 471 . 2  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
14 ralxfr2d.3 . 2  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
1512, 13, 14ralxfrd 4729 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   E.wex 1550    = wceq 1652    e. wcel 1725   A.wral 2697   E.wrex 2698
This theorem is referenced by:  rexxfr2d  4732  ralrn  5865  ralima  5970  cnrest2  17342  cnprest2  17346  consuba  17475  subislly  17536  trfbas2  17867  trfil2  17911  flimrest  18007  fclsrest  18048  tsmssubm  18164
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 2416
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950
  Copyright terms: Public domain W3C validator