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Theorem sbn 2274
Description: Negation inside and outside of substitution are equivalent. (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 1829 . . 3 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)))
2 exanali 1752 . . . 4 (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦𝜑))
32anbi2i 717 . . 3 (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑)) ↔ ((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦𝜑)))
4 annim 434 . . 3 (((𝑥 = 𝑦 → ¬ 𝜑) ∧ ¬ ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
51, 3, 43bitri 281 . 2 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
6 dfsb3 2257 . 2 ([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦 → ¬ 𝜑) → ∀𝑥(𝑥 = 𝑦𝜑)))
75, 6xchbinxr 320 1 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 191  wa 378  wal 1466  wex 1692  [wsb 1828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-10 1965  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-ex 1693  df-nf 1697  df-sb 1829
This theorem is referenced by:  sbi2  2276  sbor  2281  sban  2282  sbex  2346  sbcng  3332  difab  3738  bj-abfal  31693  wl-sb8et  32112  pm13.196a  37122
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