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Theorem cbvrexsv 2852
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 1619 . . 3  |-  F/ z
ph
2 nfs1v 2111 . . 3  |-  F/ x [ z  /  x ] ph
3 sbequ12 1924 . . 3  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
41, 2, 3cbvrex 2837 . 2  |-  ( E. x  e.  A  ph  <->  E. z  e.  A  [
z  /  x ] ph )
5 nfv 1619 . . . 4  |-  F/ y
ph
65nfsb 2114 . . 3  |-  F/ y [ z  /  x ] ph
7 nfv 1619 . . 3  |-  F/ z [ y  /  x ] ph
8 sbequ 2065 . . 3  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
96, 7, 8cbvrex 2837 . 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 1648   E.wrex 2620
This theorem is referenced by:  rspesbca  3147  ac6sf  8203  ac6gf  25735  cbvexsv  28040
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ral 2624  df-rex 2625
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