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Theorem cvbtrcl 14886
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 14885 . 2 (𝑥 = 𝑦 → ((𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)))
21cbvabv 2806 1 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  {cab 2710  wss 3914  ccom 5641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3449  df-in 3921  df-ss 3931  df-br 5110  df-opab 5172  df-co 5646
This theorem is referenced by: (None)
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