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Theorem cbvexsv 28633
Description: A theorem pertaining to the substitution for an existentially quantified variable when the substituted variable does not occur in the quantified wff. (Contributed by Alan Sare, 22-Jul-2012.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
cbvexsv  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Distinct variable group:    ph, y
Allowed substitution hint:    ph( x)

Proof of Theorem cbvexsv
StepHypRef Expression
1 cbvrexsv 2944 . 2  |-  ( E. x  e.  _V  ph  <->  E. y  e.  _V  [
y  /  x ] ph )
2 rexv 2970 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ph )
3 rexv 2970 . 2  |-  ( E. y  e.  _V  [
y  /  x ] ph 
<->  E. y [ y  /  x ] ph )
41, 2, 33bitr3i 267 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550   [wsb 1658   E.wrex 2706   _Vcvv 2956
This theorem is referenced by:  onfrALTlem1  28634  onfrALTlem1VD  29002
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  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958
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