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

Theorem rexxfrd 4606
Description: Transfer universal quantification from a variable  x to another variable  y contained in expression  A. (Contributed by FL, 10-Apr-2007.) (Revised by Mario Carneiro, 15-Aug-2014.)
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
rexxfrd  |-  ( 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)

Proof of Theorem rexxfrd
StepHypRef Expression
1 ralxfrd.1 . . . 4  |-  ( (
ph  /\  y  e.  C )  ->  A  e.  B )
2 ralxfrd.2 . . . 4  |-  ( (
ph  /\  x  e.  B )  ->  E. y  e.  C  x  =  A )
3 ralxfrd.3 . . . . 5  |-  ( (
ph  /\  x  =  A )  ->  ( ps 
<->  ch ) )
43notbid 292 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  <->  -.  ch )
)
51, 2, 4ralxfrd 4605 . . 3  |-  ( ph  ->  ( A. x  e.  B  -.  ps  <->  A. y  e.  C  -.  ch )
)
65notbid 292 . 2  |-  ( ph  ->  ( -.  A. x  e.  B  -.  ps  <->  -.  A. y  e.  C  -.  ch )
)
7 dfrex2 2855 . 2  |-  ( E. x  e.  B  ps  <->  -. 
A. x  e.  B  -.  ps )
8 dfrex2 2855 . 2  |-  ( E. y  e.  C  ch  <->  -. 
A. y  e.  C  -.  ch )
96, 7, 83bitr4g 288 1  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. y  e.  C  ch )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 367    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ral 2759  df-rex 2760  df-v 3061
This theorem is referenced by:  cmpfi  20201  elfm  20740  metucnOLD  21383  rlimcnp  23621  fargshiftfo  25055  rmoxfrdOLD  27806  rmoxfrd  27807  iunrdx  27861  dvh4dimat  34458  mapdcv  34680  elrfirn  34989
  Copyright terms: Public domain W3C validator