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Theorem cleljust 1945
Description: When the class variables in definition df-clel 2501 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1938 with the class variables in wcel 1937. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 1880 in order to remove dependencies on ax-10 1965, ax-12 1983, ax-13 2137. Note that there is no DV condition on 𝑥, 𝑦, that is, on the variables of the left-hand side. (Revised by BJ, 29-Dec-2020.)
Assertion
Ref Expression
cleljust (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljust
StepHypRef Expression
1 elequ1 1944 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
21equsexvw 1880 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
32bicomi 209 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 191  wa 378  wex 1692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939
This theorem depends on definitions:  df-bi 192  df-an 380  df-ex 1693
This theorem is referenced by:  bj-dfclel  31681
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