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Theorem equvinv 2029
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2138, ax-13 2385. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.) (Proof shortened by Wolf Lammen, 12-Jul-2022.)
Assertion
Ref Expression
equvinv (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem equvinv
StepHypRef Expression
1 equequ1 2025 . . 3 (𝑧 = 𝑥 → (𝑧 = 𝑦𝑥 = 𝑦))
21equsexvw 2004 . 2 (∃𝑧(𝑧 = 𝑥𝑧 = 𝑦) ↔ 𝑥 = 𝑦)
32bicomi 225 1 (𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wex 1773
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1774
This theorem is referenced by:  ax8  2113  ax9  2121  ax13  2388  cossid  35587
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