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

Theorem cbvrexsv 2776
Description: Change bound variable by using a substitution. (Contributed by NM, 2-Mar-2008.) (Revised by Andrew Salmon, 11-Jul-2011.)
Assertion
Ref Expression
cbvrexsv  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  [
y  /  x ] ph )
Distinct variable groups:    x, A    ph, y    y, A
Allowed substitution hint:    ph( x)

Proof of Theorem cbvrexsv
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 nfv 1605 . . 3  |-  F/ z
ph
2 nfs1v 2045 . . 3  |-  F/ x [ z  /  x ] ph
3 sbequ12 1860 . . 3  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
41, 2, 3cbvrex 2761 . 2  |-  ( E. x  e.  A  ph  <->  E. z  e.  A  [
z  /  x ] ph )
5 nfv 1605 . . . 4  |-  F/ y
ph
65nfsb 2048 . . 3  |-  F/ y [ z  /  x ] ph
7 nfv 1605 . . 3  |-  F/ z [ y  /  x ] ph
8 sbequ 2000 . . 3  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
96, 7, 8cbvrex 2761 . 2  |-  ( E. z  e.  A  [
z  /  x ] ph 
<->  E. y  e.  A  [ y  /  x ] ph )
104, 9bitri 240 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  [
y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   [wsb 1629   E.wrex 2544
This theorem is referenced by:  rspesbca  3071  ac6sf  8116  ac6gf  26411  cbvexsv  28312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549
  Copyright terms: Public domain W3C validator