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Theorem cleljust 2038
Description: When the class variables in definition df-clel 2647 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 2031 with the class variables in wcel 2030. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 1978 in order to remove dependencies on ax-10 2059, ax-12 2087, ax-13 2282. 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 2037 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
21equsexvw 1978 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
32bicomi 214 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wex 1744
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032
This theorem depends on definitions:  df-bi 197  df-an 385  df-ex 1745
This theorem is referenced by:  bj-dfclel  33014
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