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Theorem equvelv 2114
Description: A specialized version of equvel 2484 with distinct variable restrictions and fewer axiom usage. (Contributed by Wolf Lammen, 10-Apr-2021.)
Assertion
Ref Expression
equvelv (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvelv
StepHypRef Expression
1 equtrr 2104 . . 3 (𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))
21alrimiv 2004 . 2 (𝑥 = 𝑦 → ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
3 equs4v 2085 . . 3 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
4 equvinv 2112 . . 3 (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
53, 4sylibr 224 . 2 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) → 𝑥 = 𝑦)
62, 5impbii 199 1 (𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1630  wex 1853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1854
This theorem is referenced by: (None)
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