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Theorem sb3bOLD 2497
 Description: Obsolete version of sb3b 2490 as of 21-Feb-2024. (Contributed by BJ, 6-Oct-2018.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sb3bOLD (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))

Proof of Theorem sb3bOLD
StepHypRef Expression
1 sb1 2492 . 2 ([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
2 sb3 2491 . 2 (¬ ∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑥 = 𝑦𝜑) → [𝑦 / 𝑥]𝜑))
31, 2impbid2 229 1 (¬ ∀𝑥 𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑 ↔ ∃𝑥(𝑥 = 𝑦𝜑)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536  ∃wex 1781  [wsb 2069 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-12 2175  ax-13 2379 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070 This theorem is referenced by: (None)
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