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Theorem sb4v 2248
 Description: Version of sb4 2431 with a disjoint variable condition instead of a distinctor antecedent, which does not require ax-13 2333. (Contributed by BJ, 23-Jun-2019.)
Assertion
Ref Expression
sb4v ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb4v
StepHypRef Expression
1 sb1 2014 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 sb56 2247 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
31, 2sylib 210 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386  ∀wal 1599  ∃wex 1823  [wsb 2011 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-10 2134  ax-12 2162 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-ex 1824  df-nf 1828  df-sb 2012 This theorem is referenced by:  sb6  2249  sbi1v  2277  bj-hbs1  33354
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