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Theorem cleq1 14877
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 14876 . . 3 (𝑅 = 𝑆 → ((𝑅𝑟𝜑) ↔ (𝑆𝑟𝜑)))
21abbidv 2802 . 2 (𝑅 = 𝑆 → {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
32inteqd 4916 1 (𝑅 = 𝑆 {𝑟 ∣ (𝑅𝑟𝜑)} = {𝑟 ∣ (𝑆𝑟𝜑)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  {cab 2710  wss 3914   cint 4911
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-v 3449  df-in 3921  df-ss 3931  df-int 4912
This theorem is referenced by:  trcleq1  14883
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