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Theorem cbvrexsv 2695
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 1508 . . 3  |-  F/ z
ph
2 nfs1v 1919 . . 3  |-  F/ x [ z  /  x ] ph
3 sbequ12 1751 . . 3  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
41, 2, 3cbvrex 2677 . 2  |-  ( E. x  e.  A  ph  <->  E. z  e.  A  [
z  /  x ] ph )
5 nfv 1508 . . . 4  |-  F/ y
ph
65nfsb 1926 . . 3  |-  F/ y [ z  /  x ] ph
7 nfv 1508 . . 3  |-  F/ z [ y  /  x ] ph
8 sbequ 1820 . . 3  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
96, 7, 8cbvrex 2677 . 2  |-  ( E. z  e.  A  [
z  /  x ] ph 
<->  E. y  e.  A  [ y  /  x ] ph )
104, 9bitri 183 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  [
y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 104   [wsb 1742   E.wrex 2436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-cleq 2150  df-clel 2153  df-nfc 2288  df-rex 2441
This theorem is referenced by:  rspesbca  3021  rexxpf  4732
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