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

Theorem sban 2118
Description: Conjunction inside and outside of a substitution are equivalent. (Contributed by NM, 5-Aug-1993.)
Assertion
Ref Expression
sban  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )

Proof of Theorem sban
StepHypRef Expression
1 sbn 2111 . . 3  |-  ( [ y  /  x ]  -.  ( ph  ->  -.  ps )  <->  -.  [ y  /  x ] ( ph  ->  -.  ps ) )
2 sbim 2114 . . . 4  |-  ( [ y  /  x ]
( ph  ->  -.  ps ) 
<->  ( [ y  /  x ] ph  ->  [ y  /  x ]  -.  ps ) )
3 sbn 2111 . . . . 5  |-  ( [ y  /  x ]  -.  ps  <->  -.  [ y  /  x ] ps )
43imbi2i 304 . . . 4  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ]  -.  ps ) 
<->  ( [ y  /  x ] ph  ->  -.  [ y  /  x ] ps ) )
52, 4bitri 241 . . 3  |-  ( [ y  /  x ]
( ph  ->  -.  ps ) 
<->  ( [ y  /  x ] ph  ->  -.  [ y  /  x ] ps ) )
61, 5xchbinx 302 . 2  |-  ( [ y  /  x ]  -.  ( ph  ->  -.  ps )  <->  -.  ( [
y  /  x ] ph  ->  -.  [ y  /  x ] ps )
)
7 df-an 361 . . 3  |-  ( (
ph  /\  ps )  <->  -.  ( ph  ->  -.  ps ) )
87sbbii 1661 . 2  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  [ y  /  x ]  -.  ( ph  ->  -.  ps ) )
9 df-an 361 . 2  |-  ( ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) 
<->  -.  ( [ y  /  x ] ph  ->  -.  [ y  /  x ] ps ) )
106, 8, 93bitr4i 269 1  |-  ( [ y  /  x ]
( ph  /\  ps )  <->  ( [ y  /  x ] ph  /\  [ y  /  x ] ps ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   [wsb 1655
This theorem is referenced by:  sb3an  2119  sbbi  2120  sbabel  2566  cbvreu  2890  sbcan  3163  sbcang  3164  rmo3  3208  inab  3569  difab  3570  exss  4386  inopab  4964  mo5f  23925  rmo3f  23935  iuninc  23964  suppss2f  24002  fmptdF  24022  disjdsct  24043  esumpfinvalf  24419  measiuns  24524  ballotlemodife  24708  sb5ALT  28320  2uasbanh  28359  2uasbanhVD  28732  sb5ALTVD  28734
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656
  Copyright terms: Public domain W3C validator