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Theorem sbn 2118
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 1727 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  ( ( x  =  y  ->  -.  ph )  /\  E. x
( x  =  y  /\  -.  ph )
) )
2 exanali 1657 . . . 4  |-  ( E. x ( x  =  y  /\  -.  ph ) 
<->  -.  A. x ( x  =  y  ->  ph ) )
32anbi2i 694 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  E. x ( x  =  y  /\  -.  ph ) )  <->  ( (
x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph ) ) )
4 annim 425 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph )
)  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
51, 3, 43bitri 271 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
6 dfsb3 2101 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
) )
75, 6xchbinxr 311 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    /\ wa 369   A.wal 1381   E.wex 1599   [wsb 1726
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-10 1823  ax-12 1840  ax-13 1985
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-ex 1600  df-nf 1604  df-sb 1727
This theorem is referenced by:  sbi2  2120  sbor  2125  sban  2126  sbex  2193  sbcng  3354  difab  3752  wl-sb8et  29977  pm13.196a  31275  bj-abfal  34357
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