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.

Step	Hyp	Ref	Expression
1		sbnv 1823	. . . 4 |- ( [ z / x ] -. ph <-> -. [ z / x ] ph )
2	1	sbbii 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 )
4	2, 3	bitri 183	. 2 |- ( [ y / z ] [ z / x ] -. ph <-> -. [ y / z ] [ z / x ] ph )
5		ax-17 1471	. . . 4 |- ( ph -> A. z ph )
6	5	hbn 1596	. . 3 |- ( -. ph -> A. z -. ph )
7	6	sbco2v 1876	. 2 |- ( [ y / z ] [ z / x ] -. ph <-> [ y / x ] -. ph )
8	5	sbco2v 1876	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9	8	notbii 632	. 2 |- ( -. [ y / z ] [ z / x ] ph <-> -. [ y / x ] ph )
10	4, 7, 9	3bitr3i 209	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

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