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Theorem rexxfrd 4565
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 285 . . . 4  |-  ( (
ph  /\  x  =  A )  ->  ( -.  ps  <->  -.  ch )
)
51, 2, 4ralxfrd 4564 . . 3  |-  ( ph  ->  ( A. x  e.  B  -.  ps  <->  A. y  e.  C  -.  ch )
)
65notbid 285 . 2  |-  ( ph  ->  ( -.  A. x  e.  B  -.  ps  <->  -.  A. y  e.  C  -.  ch )
)
7 dfrex2 2569 . 2  |-  ( E. x  e.  B  ps  <->  -. 
A. x  e.  B  -.  ps )
8 dfrex2 2569 . 2  |-  ( E. y  e.  C  ch  <->  -. 
A. y  e.  C  -.  ch )
96, 7, 83bitr4g 279 1  |-  ( ph  ->  ( E. x  e.  B  ps  <->  E. y  e.  C  ch )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557
This theorem is referenced by:  cmpfi  17151  elfm  17658  rlimcnp  20276  rmoxfrdOLD  23162  rmoxfrd  23163  iunrdx  23177  elrfirn  26873  fargshiftfo  28383  dvh4dimat  32250  mapdcv  32472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803
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