Theorem sbn 1268

Description: Negation inside and outside of substitution are equivalent.

Assertion

Ref	Expression
sbn	|- ([y / x] -. ph <-> -. [y / x]ph)

Proof of Theorem sbn

Step	Hyp	Ref	Expression
1		sbequ2 1216	. . . . 5 |- (x = y -> ([y / x] -. ph -> -. ph))
2		sbequ2 1216	. . . . 5 |- (x = y -> ([y / x]ph -> ph))
3	1, 2	nsyld 116	. . . 4 |- (x = y -> ([y / x] -. ph -> -. [y / x]ph))
4	3	a4s 1020	. . 3 |- (A.x x = y -> ([y / x] -. ph -> -. [y / x]ph))
5		sb4 1260	. . . 4 |- (-. A.x x = y -> ([y / x] -. ph -> A.x(x = y -> -. ph)))
6		sb1 1213	. . . . . 6 |- ([y / x]ph -> E.x(x = y /\ ph))
7		equs3 1186	. . . . . 6 |- (E.x(x = y /\ ph) <-> -. A.x(x = y -> -. ph))
8	6, 7	sylib 196	. . . . 5 |- ([y / x]ph -> -. A.x(x = y -> -. ph))
9	8	con2i 97	. . . 4 |- (A.x(x = y -> -. ph) -> -. [y / x]ph)
10	5, 9	syl6 22	. . 3 |- (-. A.x x = y -> ([y / x] -. ph -> -. [y / x]ph))
11	4, 10	pm2.61i 124	. 2 |- ([y / x] -. ph -> -. [y / x]ph)
12		sbequ1 1215	. . . . . 6 |- (x = y -> (ph -> [y / x]ph))
13	12	con3d 95	. . . . 5 |- (x = y -> (-. [y / x]ph -> -. ph))
14	13	com12 11	. . . 4 |- (-. [y / x]ph -> (x = y -> -. ph))
15		sb2 1214	. . . . . . 7 |- (A.x(x = y -> -. -. ph) -> [y / x] -. -. ph)
16		notnot 159	. . . . . . . 8 |- (ph <-> -. -. ph)
17	16	sbbii 1211	. . . . . . 7 |- ([y / x]ph <-> [y / x] -. -. ph)
18	15, 17	sylibr 198	. . . . . 6 |- (A.x(x = y -> -. -. ph) -> [y / x]ph)
19	18	con3i 98	. . . . 5 |- (-. [y / x]ph -> -. A.x(x = y -> -. -. ph))
20		equs3 1186	. . . . 5 |- (E.x(x = y /\ -. ph) <-> -. A.x(x = y -> -. -. ph))
21	19, 20	sylibr 198	. . . 4 |- (-. [y / x]ph -> E.x(x = y /\ -. ph))
22	14, 21	jca 286	. . 3 |- (-. [y / x]ph -> ((x = y -> -. ph) /\ E.x(x = y /\ -. ph)))
23		df-sb 1209	. . 3 |- ([y / x] -. ph <-> ((x = y -> -. ph) /\ E.x(x = y /\ -. ph)))
24	22, 23	sylibr 198	. 2 |- (-. [y / x]ph -> [y / x] -. ph)
25	11, 24	impbii 155	1 |- ([y / x] -. ph <-> -. [y / x]ph)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990  E.wex 1016  [wsbc 1207

This theorem is referenced by:  sbi2 1270  sbor 1272  sban 1274  a4sbe 1280  sb8e 1300  sbex 1387  sbcng 2017  difab 2321

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-10 1002  ax-12 1004  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-an 223  df-ex 1017  df-sb 1209

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