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Theorem bj-sb4v 32882
Description: Version of sb4 2384 with a dv condition, which does not require ax-13 2282. (Contributed by BJ, 23-Jun-2019.) Together with bj-sb2v 32878, this allosw to remove ax-13 2282 from sb6 2457 (see bj-sb6 32892). Note that this subsumes the version of sb4b 2386 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 1940 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 sb56 2188 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
31, 2sylib 208 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  wal 1521  wex 1744  [wsb 1937
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-10 2059  ax-12 2087
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ex 1745  df-nf 1750  df-sb 1938
This theorem is referenced by:  bj-hbs1  32883
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