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Theorem sbn 2134
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 1660 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  ( ( x  =  y  ->  -.  ph )  /\  E. x
( x  =  y  /\  -.  ph )
) )
2 exanali 1596 . . . 4  |-  ( E. x ( x  =  y  /\  -.  ph ) 
<->  -.  A. x ( x  =  y  ->  ph ) )
32anbi2i 677 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  E. x ( x  =  y  /\  -.  ph ) )  <->  ( (
x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph ) ) )
4 annim 416 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph )
)  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
51, 3, 43bitri 264 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
6 dfsb3 2113 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
) )
75, 6xchbinxr 304 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 178    /\ wa 360   A.wal 1550   E.wex 1551   [wsb 1659
This theorem is referenced by:  sbi2  2137  sbor  2143  sban  2144  spsbeOLD  2152  sbex  2211  sbcng  3207  difab  3595  pm13.196a  27629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660
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