Theorem sbn 2062

Description: Negation inside and outside of substitution are equivalent. (Contributed by NM, 5-Aug-1993.)

Assertion

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

Proof of Theorem sbn

Step	Hyp	Ref	Expression
1		sbequ2 1650	. . . . 5 |- (x = y -> ([y / x] -. ph -> -. ph))
2		sbequ2 1650	. . . . 5 |- (x = y -> ([y / x]ph -> ph))
3	1, 2	nsyld 132	. . . 4 |- (x = y -> ([y / x] -. ph -> -. [y / x]ph))
4	3	sps 1754	. . 3 |- (A.x x = y -> ([y / x] -. ph -> -. [y / x]ph))
5		sb4 2053	. . . 4 |- (-. A.x x = y -> ([y / x] -. ph -> A.x(x = y -> -. ph)))
6		sb1 1651	. . . . . 6 |- ([y / x]ph -> E.x(x = y /\ ph))
7		equs3 1644	. . . . . 6 |- (E.x(x = y /\ ph) <-> -. A.x(x = y -> -. ph))
8	6, 7	sylib 188	. . . . 5 |- ([y / x]ph -> -. A.x(x = y -> -. ph))
9	8	con2i 112	. . . 4 |- (A.x(x = y -> -. ph) -> -. [y / x]ph)
10	5, 9	syl6 29	. . 3 |- (-. A.x x = y -> ([y / x] -. ph -> -. [y / x]ph))
11	4, 10	pm2.61i 156	. 2 |- ([y / x] -. ph -> -. [y / x]ph)
12		sbequ1 1918	. . . 4 |- (x = y -> (ph -> [y / x]ph))
13	12	con3rr3 128	. . 3 |- (-. [y / x]ph -> (x = y -> -. ph))
14		sb2 2023	. . . . . 6 |- (A.x(x = y -> -. -. ph) -> [y / x] -. -. ph)
15		notnot 282	. . . . . . 7 |- (ph <-> -. -. ph)
16	15	sbbii 1653	. . . . . 6 |- ([y / x]ph <-> [y / x] -. -. ph)
17	14, 16	sylibr 203	. . . . 5 |- (A.x(x = y -> -. -. ph) -> [y / x]ph)
18	17	con3i 127	. . . 4 |- (-. [y / x]ph -> -. A.x(x = y -> -. -. ph))
19		equs3 1644	. . . 4 |- (E.x(x = y /\ -. ph) <-> -. A.x(x = y -> -. -. ph))
20	18, 19	sylibr 203	. . 3 |- (-. [y / x]ph -> E.x(x = y /\ -. ph))
21		df-sb 1649	. . 3 |- ([y / x] -. ph <-> ((x = y -> -. ph) /\ E.x(x = y /\ -. ph)))
22	13, 20, 21	sylanbrc 645	. 2 |- (-. [y / x]ph -> [y / x] -. ph)
23	11, 22	impbii 180	1 |- ([y / x] -. ph <-> -. [y / x]ph)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649

This theorem is referenced by:  sbi2  2064  sbor  2066  sban  2069  spsbe  2075  sb8e  2093  sbex  2128  sbcng  3087  difab  3524  complab  3525