Theorem sbn 1926

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 1861	. . . 4 |- ( [ z / x ] -. ph <-> -. [ z / x ] ph )
2	1	sbbii 1739	. . 3 |- ( [ y / z ] [ z / x ] -. ph <-> [ y / z ] -. [ z / x ] ph )
3		sbnv 1861	. . 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 1507	. . . 4 |- ( ph -> A. z ph )
6	5	hbn 1633	. . 3 |- ( -. ph -> A. z -. ph )
7	6	sbco2vh 1919	. 2 |- ( [ y / z ] [ z / x ] -. ph <-> [ y / x ] -. ph )
8	5	sbco2vh 1919	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9	8	notbii 658	. 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 1736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737

This theorem is referenced by:  sbcng  2953  difab  3350  rabeq0  3397  abeq0  3398  ssfirab  6830