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Theorem sb4 2248
 Description: One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 14-May-1993.)
Assertion
Ref Expression
sb4 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem sb4
StepHypRef Expression
1 sb1 1833 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 equs5 2243 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑)))
31, 2syl5ib 232 1 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ∧ 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:  sb4b  2250  hbsb2  2251  dfsb2  2265  sbequi  2267  sbi1  2284
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