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Theorem sbn 1881
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 1823 . . . 4  |-  ( [ z  /  x ]  -.  ph  <->  -.  [ z  /  x ] ph )
21sbbii 1702 . . 3  |-  ( [ y  /  z ] [ z  /  x ]  -.  ph  <->  [ y  /  z ]  -.  [ z  /  x ] ph )
3 sbnv 1823 . . 3  |-  ( [ y  /  z ]  -.  [ z  /  x ] ph  <->  -.  [ y  /  z ] [
z  /  x ] ph )
42, 3bitri 183 . 2  |-  ( [ y  /  z ] [ z  /  x ]  -.  ph  <->  -.  [ y  /  z ] [
z  /  x ] ph )
5 ax-17 1471 . . . 4  |-  ( ph  ->  A. z ph )
65hbn 1596 . . 3  |-  ( -. 
ph  ->  A. z  -.  ph )
76sbco2v 1876 . 2  |-  ( [ y  /  z ] [ z  /  x ]  -.  ph  <->  [ y  /  x ]  -.  ph )
85sbco2v 1876 . . 3  |-  ( [ y  /  z ] [ z  /  x ] ph  <->  [ y  /  x ] ph )
98notbii 632 . 2  |-  ( -. 
[ y  /  z ] [ z  /  x ] ph  <->  -.  [ y  /  x ] ph )
104, 7, 93bitr3i 209 1  |-  ( [ y  /  x ]  -.  ph  <->  -.  [ y  /  x ] ph )
Colors of variables: wff set class
Syntax hints:   -. wn 3    <-> wb 104   [wsb 1699
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480
This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700
This theorem is referenced by:  sbcng  2893  difab  3284  rabeq0  3331  abeq0  3332  ssfirab  6723
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