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Theorem equvelvOLD 2133
 Description: Obsolete version of equvelv 2132 as of 12-Jul-2022. (Contributed by Wolf Lammen, 10-Apr-2021.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
equvelvOLD (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvelvOLD
StepHypRef Expression
1 equtrr 2121 . . 3 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
21alrimiv 2023 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
3 equs4v 2102 . . 3 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
4 equvinv 2129 . . 3 (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
53, 4sylibr 226 . 2 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
62, 5impbii 201 1 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385  ∀wal 1651  ∃wex 1875 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107 This theorem depends on definitions:  df-bi 199  df-an 386  df-ex 1876 This theorem is referenced by: (None)
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