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Theorem cleljust 1984
Description: When the class variables in definition df-clel 2605 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 1977 with the class variables in wcel 1976. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 1918 in order to remove dependencies on ax-10 2005, ax-12 2033, ax-13 2233. 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 1983 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
21equsexvw 1918 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
32bicomi 212 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  wex 1694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978
This theorem depends on definitions:  df-bi 195  df-an 384  df-ex 1695
This theorem is referenced by:  bj-dfclel  31878
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