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

Theorem sbft 1965
Description: Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.)
Assertion
Ref Expression
sbft  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )

Proof of Theorem sbft
StepHypRef Expression
1 sb1 1632 . . 3  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 simpr 447 . . . . 5  |-  ( ( x  =  y  /\  ph )  ->  ph )
32ax-gen 1533 . . . 4  |-  A. x
( ( x  =  y  /\  ph )  ->  ph )
4 19.23t 1796 . . . 4  |-  ( F/ x ph  ->  ( A. x ( ( x  =  y  /\  ph )  ->  ph )  <->  ( E. x ( x  =  y  /\  ph )  ->  ph ) ) )
53, 4mpbii 202 . . 3  |-  ( F/ x ph  ->  ( E. x ( x  =  y  /\  ph )  ->  ph ) )
61, 5syl5 28 . 2  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  ->  ph ) )
7 nfr 1741 . . 3  |-  ( F/ x ph  ->  ( ph  ->  A. x ph )
)
8 stdpc4 1964 . . 3  |-  ( A. x ph  ->  [ y  /  x ] ph )
97, 8syl6 29 . 2  |-  ( F/ x ph  ->  ( ph  ->  [ y  /  x ] ph ) )
106, 9impbid 183 1  |-  ( F/ x ph  ->  ( [ y  /  x ] ph  <->  ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   A.wal 1527   E.wex 1528   F/wnf 1531   [wsb 1629
This theorem is referenced by:  sbf  1966  sbctt  3053
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
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1529  df-nf 1532  df-sb 1630
  Copyright terms: Public domain W3C validator