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Theorem cleq1 14334
Description: Equality of relations implies equality of closures. (Contributed by RP, 9-May-2020.)
Assertion
Ref Expression
cleq1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
Distinct variable groups:   𝑅,𝑟   𝑆,𝑟
Allowed substitution hint:   𝜑(𝑟)

Proof of Theorem cleq1
StepHypRef Expression
1 cleq1lem 14333 . . 3 (𝑅 = 𝑆 → ((𝑅𝑟𝜑) ↔ (𝑆𝑟𝜑)))
21abbidv 2886 . 2 (𝑅 = 𝑆 → {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
32inteqd 4856 1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  {cab 2800  wss 3908   cint 4851
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 2116  ax-9 2124  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-ral 3135  df-v 3471  df-in 3915  df-ss 3925  df-int 4852
This theorem is referenced by:  trcleq1  14340
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