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Theorem sbequ2OLD 2253
Description: Obsolete version of sbequ2 2252 as of 3-Feb-2024. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.) Revise df-sb 2071. (Revised by BJ, 22-Dec-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sbequ2OLD (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑𝜑))

Proof of Theorem sbequ2OLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 equvinva 2038 . 2 (𝑥 = 𝑡 → ∃𝑦(𝑥 = 𝑦𝑡 = 𝑦))
2 df-sb 2071 . . . 4 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
3 equcomi 2025 . . . . . . 7 (𝑡 = 𝑦𝑦 = 𝑡)
4 sp 2184 . . . . . . 7 (∀𝑥(𝑥 = 𝑦𝜑) → (𝑥 = 𝑦𝜑))
53, 4imim12i 62 . . . . . 6 ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → (𝑡 = 𝑦 → (𝑥 = 𝑦𝜑)))
65impcomd 415 . . . . 5 ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → ((𝑥 = 𝑦𝑡 = 𝑦) → 𝜑))
76alimi 1813 . . . 4 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) → ∀𝑦((𝑥 = 𝑦𝑡 = 𝑦) → 𝜑))
82, 7sylbi 220 . . 3 ([𝑡 / 𝑥]𝜑 → ∀𝑦((𝑥 = 𝑦𝑡 = 𝑦) → 𝜑))
9 19.23v 1944 . . 3 (∀𝑦((𝑥 = 𝑦𝑡 = 𝑦) → 𝜑) ↔ (∃𝑦(𝑥 = 𝑦𝑡 = 𝑦) → 𝜑))
108, 9sylib 221 . 2 ([𝑡 / 𝑥]𝜑 → (∃𝑦(𝑥 = 𝑦𝑡 = 𝑦) → 𝜑))
111, 10syl5com 31 1 (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wal 1536  wex 1781  [wsb 2070
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 1912  ax-6 1971  ax-7 2016  ax-12 2179
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2071
This theorem is referenced by: (None)
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