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Theorem cbvrexsv 2998
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 1764 . . 3  |-  F/ z
ph
2 nfs1v 2266 . . 3  |-  F/ x [ z  /  x ] ph
3 sbequ12 2083 . . 3  |-  ( x  =  z  ->  ( ph 
<->  [ z  /  x ] ph ) )
41, 2, 3cbvrex 2983 . 2  |-  ( E. x  e.  A  ph  <->  E. z  e.  A  [
z  /  x ] ph )
5 nfv 1764 . . . 4  |-  F/ y
ph
65nfsb 2269 . . 3  |-  F/ y [ z  /  x ] ph
7 nfv 1764 . . 3  |-  F/ z [ y  /  x ] ph
8 sbequ 2205 . . 3  |-  ( z  =  y  ->  ( [ z  /  x ] ph  <->  [ y  /  x ] ph ) )
96, 7, 8cbvrex 2983 . 2  |-  ( E. z  e.  A  [
z  /  x ] ph 
<->  E. y  e.  A  [ y  /  x ] ph )
104, 9bitri 257 1  |-  ( E. x  e.  A  ph  <->  E. y  e.  A  [
y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189   [wsb 1800   E.wrex 2737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1667  df-nf 1671  df-sb 1801  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ral 2741  df-rex 2742
This theorem is referenced by:  rspesbca  3315  ac6sf  8905  ac6gf  32060  cbvexsv  36913
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