MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb8e Structured version   Visualization version   Unicode version

Theorem sb8e 2256
Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
Hypothesis
Ref Expression
sb5rf.1  |-  F/ y
ph
Assertion
Ref Expression
sb8e  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )

Proof of Theorem sb8e
StepHypRef Expression
1 sb5rf.1 . 2  |-  F/ y
ph
21nfs1 2196 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 2085 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbvex 2117 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188   E.wex 1665   F/wnf 1669   [wsb 1799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093
This theorem depends on definitions:  df-bi 189  df-an 373  df-ex 1666  df-nf 1670  df-sb 1800
This theorem is referenced by:  sbnf2  2270  2sb8e  2298  sb8mo  2335  mo3  2338  bnj985  29776  bj-mo3OLD  31459  sbcexf  32365  exlimddvfi  32374  pm11.58  36751
  Copyright terms: Public domain W3C validator