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Theorem sbn 2522
Description: Negation inside and outside of substitution are equivalent. For a version requiring disjoint variables, but fewer axioms, see sbnv 2334. (Contributed by NM, 14-May-1993.) (Proof shortened by Wolf Lammen, 30-Apr-2018.)
Assertion
Ref Expression
sbn ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)

Proof of Theorem sbn
StepHypRef Expression
1 df-sb 2068 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)))
2 exanali 1959 . . . 4 (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦𝜑))
32anbi2i 616 . . 3 (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦𝜑)))
4 annim 394 . . 3 (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
51, 3, 43bitri 289 . 2 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
6 dfsb3 2505 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
75, 6xchbinxr 327 1 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wal 1654  wex 1878  [wsb 2067
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-12 2220  ax-13 2389
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-ex 1879  df-nf 1883  df-sb 2068
This theorem is referenced by:  sbi2  2524  sbor  2529  sban  2530  sbex  2597  sbcng  3703  difab  4127  bj-ab0  33422  wl-sb8et  33878  pm13.196a  39453
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