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Theorem sbnvOLD 2287
 Description: Obsolete version of sbn 2253 as of 8-Jul-2023. Substitution is not affected by negation. Version of sbn 2253 with a disjoint variable condition, not requiring ax-13 2344. (Contributed by Wolf Lammen, 18-Jan-2023.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbnvOLD ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sbnvOLD
StepHypRef Expression
1 exanali 1840 . 2 (∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦𝜑))
2 sb5 2240 . 2 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ∃𝑥(𝑥 = 𝑦 ∧ ¬ 𝜑))
3 sb6 2066 . . 3 ([𝑦 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑦𝜑))
43notbii 321 . 2 (¬ [𝑦 / 𝑥]𝜑 ↔ ¬ ∀𝑥(𝑥 = 𝑦𝜑))
51, 2, 43bitr4i 304 1 ([𝑦 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑦 / 𝑥]𝜑)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396  ∀wal 1520  ∃wex 1761  [wsb 2042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043 This theorem is referenced by:  sbi2vOLD  2289
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