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Theorem cleq1 14331
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 14330 . . 3 (𝑅 = 𝑆 → ((𝑅𝑟𝜑) ↔ (𝑆𝑟𝜑)))
21abbidv 2882 . 2 (𝑅 = 𝑆 → {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
32inteqd 4872 1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  {cab 2796  wss 3933   cint 4867
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-ral 3140  df-in 3940  df-ss 3949  df-int 4868
This theorem is referenced by:  trcleq1  14337
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