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Theorem cvbtrcl 13579
 Description: Change of bound variable in class of all transitive relations which are supersets of a relation. (Contributed by RP, 5-May-2020.)
Assertion
Ref Expression
cvbtrcl {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)}
Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem cvbtrcl
StepHypRef Expression
1 trcleq2lem 13578 . 2 (𝑥 = 𝑦 → ((𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)))
21cbvabv 2734 1 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  {cab 2596   ⊆ wss 3540   ∘ ccom 5042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-in 3547  df-ss 3554  df-br 4584  df-opab 4644  df-co 5047 This theorem is referenced by: (None)
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