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

Theorem sbn 2220
Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)
Assertion
Ref Expression
sbn  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )

Proof of Theorem sbn
StepHypRef Expression
1 df-sb 1798 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  ( ( x  =  y  ->  -.  ph )  /\  E. x
( x  =  y  /\  -.  ph )
) )
2 exanali 1721 . . . 4  |-  ( E. x ( x  =  y  /\  -.  ph ) 
<->  -.  A. x ( x  =  y  ->  ph ) )
32anbi2i 700 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  E. x ( x  =  y  /\  -.  ph ) )  <->  ( (
x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph ) ) )
4 annim 427 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph )
)  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
51, 3, 43bitri 275 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
6 dfsb3 2203 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
) )
75, 6xchbinxr 313 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1442   E.wex 1663   [wsb 1797
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-12 1933  ax-13 2091
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-ex 1664  df-nf 1668  df-sb 1798
This theorem is referenced by:  sbi2  2222  sbor  2227  sban  2228  sbex  2292  sbcng  3308  difab  3712  bj-abfal  31509  wl-sb8et  31881  pm13.196a  36765
  Copyright terms: Public domain W3C validator