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

Theorem cbvexd 1988
Description: Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 2069. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.)
Hypotheses
Ref Expression
cbvald.1  |-  F/ y
ph
cbvald.2  |-  ( ph  ->  F/ y ps )
cbvald.3  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
Assertion
Ref Expression
cbvexd  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Distinct variable groups:    ph, x    ch, x
Allowed substitution hints:    ph( y)    ps( x, y)    ch( y)

Proof of Theorem cbvexd
StepHypRef Expression
1 cbvald.1 . . . 4  |-  F/ y
ph
2 cbvald.2 . . . . 5  |-  ( ph  ->  F/ y ps )
32nfnd 1809 . . . 4  |-  ( ph  ->  F/ y  -.  ps )
4 cbvald.3 . . . . 5  |-  ( ph  ->  ( x  =  y  ->  ( ps  <->  ch )
) )
5 notbi 287 . . . . 5  |-  ( ( ps  <->  ch )  <->  ( -.  ps 
<->  -.  ch ) )
64, 5syl6ib 218 . . . 4  |-  ( ph  ->  ( x  =  y  ->  ( -.  ps  <->  -. 
ch ) ) )
71, 3, 6cbvald 1986 . . 3  |-  ( ph  ->  ( A. x  -.  ps 
<-> 
A. y  -.  ch ) )
87notbid 286 . 2  |-  ( ph  ->  ( -.  A. x  -.  ps  <->  -.  A. y  -.  ch ) )
9 df-ex 1551 . 2  |-  ( E. x ps  <->  -.  A. x  -.  ps )
10 df-ex 1551 . 2  |-  ( E. y ch  <->  -.  A. y  -.  ch )
118, 9, 103bitr4g 280 1  |-  ( ph  ->  ( E. x ps  <->  E. y ch ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177   A.wal 1549   E.wex 1550   F/wnf 1553
This theorem is referenced by:  cbvexdva  1995  vtoclgft  2994  dfid3  4491  axrepndlem2  8460  axunnd  8463  axpowndlem2  8465  axpownd  8468  axregndlem2  8470  axinfndlem1  8472  axacndlem4  8477
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
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554
  Copyright terms: Public domain W3C validator