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Theorem cleq1 13923
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 13922 . . 3 (𝑅 = 𝑆 → ((𝑅𝑟𝜑) ↔ (𝑆𝑟𝜑)))
21abbidv 2879 . 2 (𝑅 = 𝑆 → {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
32inteqd 4632 1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1632  {cab 2746  wss 3715   cint 4627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-in 3722  df-ss 3729  df-int 4628
This theorem is referenced by:  trcleq1  13929
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