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

Theorem sbrim 2007
Description: Substitution with a variable not free in antecedent affects only the consequent. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 4-Oct-2016.)
Hypothesis
Ref Expression
sbrim.1  |-  F/ x ph
Assertion
Ref Expression
sbrim  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )

Proof of Theorem sbrim
StepHypRef Expression
1 sbim 2005 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 sbrim.1 . . . 4  |-  F/ x ph
32sbf 1966 . . 3  |-  ( [ y  /  x ] ph 
<-> 
ph )
43imbi1i 315 . 2  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
51, 4bitri 240 1  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  (
ph  ->  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   F/wnf 1531   [wsb 1629
This theorem is referenced by:  sbco2d  2027  2mos  2222
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-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630
  Copyright terms: Public domain W3C validator