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Theorem indifbi 30295
Description: Two ways to express equality relative to a class 𝐴. (Contributed by Thierry Arnoux, 23-Jun-2024.)
Assertion
Ref Expression
indifbi ((𝐴𝐵) = (𝐴𝐶) ↔ (𝐴𝐵) = (𝐴𝐶))

Proof of Theorem indifbi
StepHypRef Expression
1 inss1 4158 . . 3 (𝐴𝐵) ⊆ 𝐴
2 inss1 4158 . . 3 (𝐴𝐶) ⊆ 𝐴
3 rcompleq 4223 . . 3 (((𝐴𝐵) ⊆ 𝐴 ∧ (𝐴𝐶) ⊆ 𝐴) → ((𝐴𝐵) = (𝐴𝐶) ↔ (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴𝐶))))
41, 2, 3mp2an 691 . 2 ((𝐴𝐵) = (𝐴𝐶) ↔ (𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴𝐶)))
5 difin 4191 . . 3 (𝐴 ∖ (𝐴𝐵)) = (𝐴𝐵)
6 difin 4191 . . 3 (𝐴 ∖ (𝐴𝐶)) = (𝐴𝐶)
75, 6eqeq12i 2816 . 2 ((𝐴 ∖ (𝐴𝐵)) = (𝐴 ∖ (𝐴𝐶)) ↔ (𝐴𝐵) = (𝐴𝐶))
84, 7bitri 278 1 ((𝐴𝐵) = (𝐴𝐶) ↔ (𝐴𝐵) = (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  cdif 3881  cin 3883  wss 3884
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 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-rab 3118  df-v 3446  df-dif 3887  df-in 3891  df-ss 3901
This theorem is referenced by:  fressupp  30451
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