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

Theorem sbelx 2297
Description: Elimination of substitution. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sbelx  |-  ( ph  <->  E. x ( x  =  y  /\  [ x  /  y ] ph ) )
Distinct variable groups:    x, y    ph, x
Allowed substitution hint:    ph( y)

Proof of Theorem sbelx
StepHypRef Expression
1 sbid2v 2296 . 2  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
2 sb5 2269 . 2  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  E. x ( x  =  y  /\  [
x  /  y ]
ph ) )
31, 2bitr3i 259 1  |-  ( ph  <->  E. x ( x  =  y  /\  [ x  /  y ] ph ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 375   E.wex 1673   [wsb 1807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-10 1925  ax-12 1943  ax-13 2101
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-ex 1674  df-nf 1678  df-sb 1808
This theorem is referenced by:  pm13.196a  36808
  Copyright terms: Public domain W3C validator