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Theorem sbn 1963
Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 3-Feb-2018.)
Assertion
Ref Expression
sbn  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )

Proof of Theorem sbn
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 sbnv 1899 . . . 4  |-  ( [ z  /  x ]  -.  ph  <->  -.  [ z  /  x ] ph )
21sbbii 1775 . . 3  |-  ( [ y  /  z ] [ z  /  x ]  -.  ph  <->  [ y  /  z ]  -.  [ z  /  x ] ph )
3 sbnv 1899 . . 3  |-  ( [ y  /  z ]  -.  [ z  /  x ] ph  <->  -.  [ y  /  z ] [
z  /  x ] ph )
42, 3bitri 184 . 2  |-  ( [ y  /  z ] [ z  /  x ]  -.  ph  <->  -.  [ y  /  z ] [
z  /  x ] ph )
5 ax-17 1536 . . . 4  |-  ( ph  ->  A. z ph )
65hbn 1664 . . 3  |-  ( -. 
ph  ->  A. z  -.  ph )
76sbco2vh 1956 . 2  |-  ( [ y  /  z ] [ z  /  x ]  -.  ph  <->  [ y  /  x ]  -.  ph )
85sbco2vh 1956 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
98notbii 669 . 2  |-  ( -. 
[ y  /  z ] [ z  /  x ] ph  <->  -.  [ y  /  x ] ph )
104, 7, 93bitr3i 210 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 105   [wsb 1772
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773
This theorem is referenced by:  sbcng  3017  difab  3418  rabeq0  3466  abeq0  3467  ssfirab  6950
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