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Theorem sbn 2240
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 1806 . . 3  |-  ( [ y  /  x ]  -.  ph  <->  ( ( x  =  y  ->  -.  ph )  /\  E. x
( x  =  y  /\  -.  ph )
) )
2 exanali 1729 . . . 4  |-  ( E. x ( x  =  y  /\  -.  ph ) 
<->  -.  A. x ( x  =  y  ->  ph ) )
32anbi2i 708 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  E. x ( x  =  y  /\  -.  ph ) )  <->  ( (
x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph ) ) )
4 annim 432 . . 3  |-  ( ( ( x  =  y  ->  -.  ph )  /\  -.  A. x ( x  =  y  ->  ph )
)  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
51, 3, 43bitri 279 . 2  |-  ( [ y  /  x ]  -.  ph  <->  -.  ( (
x  =  y  ->  -.  ph )  ->  A. x
( x  =  y  ->  ph ) ) )
6 dfsb3 2223 . 2  |-  ( [ y  /  x ] ph 
<->  ( ( x  =  y  ->  -.  ph )  ->  A. x ( x  =  y  ->  ph )
) )
75, 6xchbinxr 318 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376   A.wal 1450   E.wex 1671   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by:  sbi2  2242  sbor  2247  sban  2248  sbex  2312  sbcng  3296  difab  3703  bj-abfal  31577  wl-sb8et  31951  pm13.196a  36835
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