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Theorem rexxfrd 4495
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 294 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  <->  -.  ch )
)
51, 2, 4ralxfrd 4494 . . 3  |-  ( ph  ->  ( A. x  e.  B  -.  ps  <->  A. y  e.  C  -.  ch )
)
65notbid 294 . 2  |-  ( ph  ->  ( -.  A. x  e.  B  -.  ps  <->  -.  A. y  e.  C  -.  ch )
)
7 dfrex2 2718 . 2  |-  ( E. x  e.  B  ps  <->  -. 
A. x  e.  B  -.  ps )
8 dfrex2 2718 . 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 369    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ral 2710  df-rex 2711  df-v 2964
This theorem is referenced by:  cmpfi  18853  elfm  19362  metucnOLD  20005  metucn  20006  rlimcnp  22244  fargshiftfo  23347  rmoxfrdOLD  25700  rmoxfrd  25701  iunrdx  25738  elrfirn  28876  dvh4dimat  34656  mapdcv  34878
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