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Theorem sbelx 2203
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 2202 . 2  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  ph )
2 sb5 2178 . 2  |-  ( [ y  /  x ] [ x  /  y ] ph  <->  E. x ( x  =  y  /\  [
x  /  y ]
ph ) )
31, 2bitr3i 244 1  |-  ( ph  <->  E. x ( x  =  y  /\  [ x  /  y ] ph ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1551   [wsb 1659
This theorem is referenced by:  sbel2x  2204  pm13.196a  27605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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