Theorem sbn 2062

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

Assertion

Ref	Expression
sbn	⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)

Proof of Theorem sbn

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

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540  ∃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