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Theorem cleljust 1855
 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 1435 with the class variables in wcel 1434. (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 1460 . . 3 (𝑥𝑦 → ∀𝑧 𝑥𝑦)
2 elequ1 1641 . . 3 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
31, 2equsex 1657 . 2 (∃𝑧(𝑧 = 𝑥𝑧𝑦) ↔ 𝑥𝑦)
43bicomi 130 1 (𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))
 Colors of variables: wff set class Syntax hints:   ∧ wa 102   ↔ wb 103  ∃wex 1422 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-4 1441  ax-13 1445  ax-17 1460  ax-i9 1464  ax-ial 1468 This theorem depends on definitions:  df-bi 115 This theorem is referenced by: (None)
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