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

Theorem sbn 2117
Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-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 1659 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  ( ( x  =  y  ->  -.  ph )  /\  E. x
( x  =  y  /\  -.  ph )
) )
2 exanali 1595 . . . 4  |-  ( E. x ( x  =  y  /\  -.  ph ) 
<->  -.  A. x ( x  =  y  ->  ph ) )
32anbi2i 676 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  E. x ( x  =  y  /\  -.  ph ) )  <->  ( (
x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph ) ) )
4 annim 415 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph )
)  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
51, 3, 43bitri 263 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
6 dfsb3 2116 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
) )
75, 6xchbinxr 303 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550   [wsb 1658
This theorem is referenced by:  sbi2  2120  sbor  2140  sban  2143  spsbeOLD  2149  sbex  2204  sbcng  3188  difab  3597  pm13.196a  27524
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659
  Copyright terms: Public domain W3C validator