Theorem sbn 1231

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 1179	. . . . 5 |- (x = y -> ([y / x] -. ph -> -. ph))
2		sbequ2 1179	. . . . 5 |- (x = y -> ([y / x]ph -> ph))
3	1, 2	nsyld 117	. . . 4 |- (x = y -> ([y / x] -. ph -> -. [y / x]ph))
4	3	a4s 984	. . 3 |- (A.x x = y -> ([y / x] -. ph -> -. [y / x]ph))
5		sb4 1223	. . . 4 |- (-. A.x x = y -> ([y / x] -. ph -> A.x(x = y -> -. ph)))
6		sb1 1176	. . . . . 6 |- ([y / x]ph -> E.x(x = y /\ ph))
7		equs3 1149	. . . . . 6 |- (E.x(x = y /\ ph) <-> -. A.x(x = y -> -. ph))
8	6, 7	sylib 198	. . . . 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 126	. 2 |- ([y / x] -. ph -> -. [y / x]ph)
12		sbequ1 1178	. . . . . 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 1177	. . . . . . 7 |- (A.x(x = y -> -. -. ph) -> [y / x] -. -. ph)
16		pm4.13 161	. . . . . . . 8 |- (ph <-> -. -. ph)
17	16	sbbii 1174	. . . . . . 7 |- ([y / x]ph <-> [y / x] -. -. ph)
18	15, 17	sylibr 200	. . . . . 6 |- (A.x(x = y -> -. -. ph) -> [y / x]ph)
19	18	con3i 98	. . . . 5 |- (-. [y / x]ph -> -. A.x(x = y -> -. -. ph))
20		equs3 1149	. . . . 5 |- (E.x(x = y /\ -. ph) <-> -. A.x(x = y -> -. -. ph))
21	19, 20	sylibr 200	. . . 4 |- (-. [y / x]ph -> E.x(x = y /\ -. ph))
22	14, 21	jca 288	. . 3 |- (-. [y / x]ph -> ((x = y -> -. ph) /\ E.x(x = y /\ -. ph)))
23		df-sb 1172	. . 3 |- ([y / x] -. ph <-> ((x = y -> -. ph) /\ E.x(x = y /\ -. ph)))
24	22, 23	sylibr 200	. 2 |- (-. [y / x]ph -> [y / x] -. ph)
25	11, 24	impbi 157	1 |- ([y / x] -. ph <-> -. [y / x]ph)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956  E.wex 980  [wsbc 1170

This theorem is referenced by:  sbi2 1233  sbor 1235  sban 1237  a4sbe 1243  sb8e 1262  sbex 1348  sbcng 1969  difab 2269

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-10 966  ax-12 968  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 981  df-sb 1172

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