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Theorem cvbtrcl 13665
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 13664 . 2 (𝑥 = 𝑦 → ((𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)))
21cbvabv 2744 1 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1480  {cab 2607  wss 3555  ccom 5078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-in 3562  df-ss 3569  df-br 4614  df-opab 4674  df-co 5083
This theorem is referenced by: (None)
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