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

Theorem sb8e 2169
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 2100 . 2  |-  F/ x [ y  /  x ] ph
3 sbequ12 1944 . 2  |-  ( x  =  y  ->  ( ph 
<->  [ y  /  x ] ph ) )
41, 2, 3cbvex 1983 1  |-  ( E. x ph  <->  E. y [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550   F/wnf 1553   [wsb 1658
This theorem is referenced by:  exsbOLD  2208  sb8mo  2300  pm11.58  27566  bnj985  29324
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
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
  Copyright terms: Public domain W3C validator