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Theorem cvbtrcl 14345
 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 14344 . 2 (𝑥 = 𝑦 → ((𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)))
21cbvabv 2866 1 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)}
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 399   = wceq 1538  {cab 2776   ⊆ wss 3881   ∘ ccom 5523 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-in 3888  df-ss 3898  df-br 5031  df-opab 5093  df-co 5528 This theorem is referenced by: (None)
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