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Theorem bj-sb4v 31945
 Description: Version of sb4 2344 with a dv condition, which does not require ax-13 2234. (Contributed by BJ, 23-Jun-2019.) Together with bj-sb2v 31941, this allosw to remove ax-13 2234 from sb6 2417 (see bj-sb6 31955). Note that this subsumes the version of sb4b 2346 with a dv condition. (Proof modification is discouraged.)
Assertion
Ref Expression
bj-sb4v ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem bj-sb4v
StepHypRef Expression
1 sb1 1870 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 sb56 2136 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
31, 2sylib 207 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ∃wex 1695  [wsb 1867 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-12 2034 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ex 1696  df-nf 1701  df-sb 1868 This theorem is referenced by:  bj-hbs1  31946
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