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Theorem bj-sb3b 31833
Description: Simplified definition of substitution when variables are distinct. This is to sb3 2247 what sb4b 2250 is to sb4 2248. Actually, one may keep only bj-sb3b 31833 and sb4b 2250 in the database, renaming them sb3 and sb4. (Contributed by BJ, 6-Oct-2018.)
Assertion
Ref Expression
bj-sb3b (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem bj-sb3b
StepHypRef Expression
1 sb1 1833 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 sb3 2247 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
31, 2impbid2 214 1 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  wal 1472  wex 1694  [wsb 1830
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-10 1966  ax-12 1983  ax-13 2137
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695  df-nf 1699  df-sb 1831
This theorem is referenced by: (None)
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