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Theorem sb4vOLDOLD 2467
 Description: Obsolete version of sb4vOLD 2067 as of 8-Jul-2023. Version of sb4OLD 2460 with a disjoint variable condition instead of a distinctor antecedent, which does not require ax-13 2344. (Contributed by BJ, 23-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sb4vOLDOLD ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem sb4vOLDOLD
StepHypRef Expression
1 sb1 2461 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 sb56 2241 . 2 (∃𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑦𝜑))
31, 2sylib 219 1 ([𝑦 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1520  ∃wex 1761  [wsb 2042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-10 2112  ax-12 2141  ax-13 2344 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-ex 1762  df-nf 1766  df-sb 2043 This theorem is referenced by: (None)
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