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Theorem sbalexOLD 2240
Description: Obsolete version of sbalex 2239 as of 14-Aug-2025. (Contributed by NM, 14-Apr-2008.) (Revised by BJ, 20-Dec-2020.) (Revised by BJ, 21-Sep-2024.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sbalexOLD (∃𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Distinct variable group:   𝑥,𝑡
Allowed substitution hints:   𝜑(𝑥,𝑡)

Proof of Theorem sbalexOLD
StepHypRef Expression
1 nfa1 2148 . . 3 𝑥𝑥(𝑥 = 𝑡𝜑)
2 ax12v2 2176 . . . 4 (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡𝜑)))
32imp 406 . . 3 ((𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
41, 3exlimi 2214 . 2 (∃𝑥(𝑥 = 𝑡𝜑) → ∀𝑥(𝑥 = 𝑡𝜑))
5 equs4v 1996 . 2 (∀𝑥(𝑥 = 𝑡𝜑) → ∃𝑥(𝑥 = 𝑡𝜑))
64, 5impbii 209 1 (∃𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1534  wex 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-10 2138  ax-12 2174
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1776  df-nf 1780
This theorem is referenced by: (None)
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