Theorem sbn 1980

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 1912	. . . 4 |- ( [ z / x ] -. ph <-> -. [ z / x ] ph )
2	1	sbbii 1788	. . 3 |- ( [ y / z ] [ z / x ] -. ph <-> [ y / z ] -. [ z / x ] ph )
3		sbnv 1912	. . 3 |- ( [ y / z ] -. [ z / x ] ph <-> -. [ y / z ] [ z / x ] ph )
4	2, 3	bitri 184	. 2 |- ( [ y / z ] [ z / x ] -. ph <-> -. [ y / z ] [ z / x ] ph )
5		ax-17 1549	. . . 4 |- ( ph -> A. z ph )
6	5	hbn 1677	. . 3 |- ( -. ph -> A. z -. ph )
7	6	sbco2vh 1973	. 2 |- ( [ y / z ] [ z / x ] -. ph <-> [ y / x ] -. ph )
8	5	sbco2vh 1973	. . 3 |- ( [ y / z ] [ z / x ] ph <-> [ y / x ] ph )
9	8	notbii 670	. 2 |- ( -. [ y / z ] [ z / x ] ph <-> -. [ y / x ] ph )
10	4, 7, 9	3bitr3i 210	1 |- ( [ y / x ] -. ph <-> -. [ y / x ] ph )

Syntax hints:   -. wn 3   <-> wb 105   [wsb 1785

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786

This theorem is referenced by:  sbcng  3039  difab  3442  rabeq0  3490  abeq0  3491  ssfirab  7035