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Theorem equvelv 2132
Description: A biconditional form of equvel 2463 with disjoint variable conditions and proved from Tarski's FOL axiom schemes. (Contributed by Andrew Salmon, 2-Jun-2011.) Reduce axiom usage. (Revised by Wolf Lammen, 10-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)
Assertion
Ref Expression
equvelv (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ 𝑥 = 𝑦)
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvelv
StepHypRef Expression
1 equequ1 2124 . 2 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
21equsalvw 2103 1 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ 𝑥 = 𝑦)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wal 1651
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:  eu6  2613  eu6OLD  2614
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