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Theorem cleljust 2131
 Description: When the class variables of set theory 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 2126 with the class variables in wcel 2125. (Contributed by NM, 28-Jan-2004.)
Assertion
Ref Expression
cleljust (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljust
StepHypRef Expression
1 ax-17 1503 . . 3 (𝑥𝑦 → ∀𝑧 𝑥𝑦)
2 elequ1 2129 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
31, 2equsex 1705 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
43bicomi 131 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
 Colors of variables: wff set class Syntax hints:   ∧ wa 103   ↔ wb 104  ∃wex 1469 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1424  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-13 2127 This theorem depends on definitions:  df-bi 116 This theorem is referenced by: (None)
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