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Theorem equvelv 2039
 Description: A biconditional form of equvel 2481 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 2033 . 2 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
21equsalvw 2011 1 (∀𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ 𝑥 = 𝑦)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536 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 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782 This theorem is referenced by:  eu6lem  2659
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