Theorem sbn 1867

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.

Step	Hyp	Ref	Expression
1		sbnv 1809	. . . 4 |- ( [ z / x ] -. ph <-> -. [ z / x ] ph )
2	1	sbbii 1688	. . 3 |- ( [ y / z ] [ z / x ] -. ph <-> [ y / z ] -. [ z / x ] ph )
3		sbnv 1809	. . 3 |- ( [ y / z ] -. [ z / x ] ph <-> -. [ y / z ] [ z / x ] ph )
4	2, 3	bitri 182	. 2 |- ( [ y / z ] [ z / x ] -. ph <-> -. [ y / z ] [ z / x ] ph )
5		ax-17 1459	. . . 4 |- ( ph -> A. z ph )
6	5	hbn 1584	. . 3 |- ( -. ph -> A. z -. ph )
7	6	sbco2v 1862	. 2 |- ( [ y / z ] [ z / x ] -. ph <-> [ y / x ] -. ph )
8	5	sbco2v 1862	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9	8	notbii 626	. 2 |- ( -. [ y / z ] [ z / x ] ph <-> -. [ y / x ] ph )
10	4, 7, 9	3bitr3i 208	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Syntax hints:   -. wn 3   <-> wb 103   [wsb 1685

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686

This theorem is referenced by:  sbcng  2854  difab  3233  rabeq0  3274  abeq0  3275