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Theorem cbvexsv 28312
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 2776 . 2  |-  ( E. x  e.  _V  ph  <->  E. y  e.  _V  [
y  /  x ] ph )
2 rexv 2802 . 2  |-  ( E. x  e.  _V  ph  <->  E. x ph )
3 rexv 2802 . 2  |-  ( E. y  e.  _V  [
y  /  x ] ph 
<->  E. y [ y  /  x ] ph )
41, 2, 33bitr3i 266 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1528   [wsb 1629   E.wrex 2544   _Vcvv 2788
This theorem is referenced by:  onfrALTlem1  28313  onfrALTlem1VD  28666
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-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790
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