Theorem sbn 1826

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 1768	. . . 4 |- ( [ z / x ] -. ph <-> -. [ z / x ] ph )
2	1	sbbii 1648	. . 3 |- ( [ y / z ] [ z / x ] -. ph <-> [ y / z ] -. [ z / x ] ph )
3		sbnv 1768	. . 3 |- ( [ y / z ] -. [ z / x ] ph <-> -. [ y / z ] [ z / x ] ph )
4	2, 3	bitri 173	. 2 |- ( [ y / z ] [ z / x ] -. ph <-> -. [ y / z ] [ z / x ] ph )
5		ax-17 1419	. . . 4 |- ( ph -> A. z ph )
6	5	hbn 1544	. . 3 |- ( -. ph -> A. z -. ph )
7	6	sbco2v 1821	. 2 |- ( [ y / z ] [ z / x ] -. ph <-> [ y / x ] -. ph )
8	5	sbco2v 1821	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9	8	notbii 594	. 2 |- ( -. [ y / z ] [ z / x ] ph <-> -. [ y / x ] ph )
10	4, 7, 9	3bitr3i 199	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Syntax hints:   -. wn 3   <-> wb 98   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646

This theorem is referenced by:  sbcng  2803  difab  3206  rabeq0  3247  abeq0  3248