Theorem sbn 1823

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 1765	. . . 4 |- ([z / x] -. ph <-> -. [z / x]ph)
2	1	sbbii 1645	. . 3 |- ([y / z][z / x] -. ph <-> [y / z] -. [z / x]ph)
3		sbnv 1765	. . 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 1416	. . . 4 |- (ph -> A.zph)
6	5	hbn 1541	. . 3 |- (-. ph -> A.z -. ph)
7	6	sbco2v 1818	. 2 |- ([y / z][z / x] -. ph <-> [y / x] -. ph)
8	5	sbco2v 1818	. . 3 |- ([y / z][z / x]ph <-> [y / x]ph)
9	8	notbii 593	. 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 1642

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643

This theorem is referenced by:  sbcng  2797  difab  3200  rabeq0  3241  abeq0  3242