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Theorem cvbtrcl 13941
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 13940 . 2 (𝑥 = 𝑦 → ((𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥) ↔ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)))
21cbvabv 2896 1 {𝑥 ∣ (𝑅𝑥 ∧ (𝑥𝑥) ⊆ 𝑥)} = {𝑦 ∣ (𝑅𝑦 ∧ (𝑦𝑦) ⊆ 𝑦)}
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  {cab 2757  wss 3723  ccom 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-in 3730  df-ss 3737  df-br 4787  df-opab 4847  df-co 5258
This theorem is referenced by: (None)
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