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| Mirrors > Home > MPE Home > Th. List > Mathboxes > indifbi | Structured version Visualization version GIF version | ||
| Description: Two ways to express equality relative to a class 𝐴. (Contributed by Thierry Arnoux, 23-Jun-2024.) | 
| Ref | Expression | 
|---|---|
| indifbi | ⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝐶)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | inss1 4236 | . . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴 | |
| 2 | inss1 4236 | . . 3 ⊢ (𝐴 ∩ 𝐶) ⊆ 𝐴 | |
| 3 | rcompleq 4304 | . . 3 ⊢ (((𝐴 ∩ 𝐵) ⊆ 𝐴 ∧ (𝐴 ∩ 𝐶) ⊆ 𝐴) → ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∩ 𝐶)))) | |
| 4 | 1, 2, 3 | mp2an 692 | . 2 ⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∩ 𝐶))) | 
| 5 | difin 4271 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ 𝐵) | |
| 6 | difin 4271 | . . 3 ⊢ (𝐴 ∖ (𝐴 ∩ 𝐶)) = (𝐴 ∖ 𝐶) | |
| 7 | 5, 6 | eqeq12i 2754 | . 2 ⊢ ((𝐴 ∖ (𝐴 ∩ 𝐵)) = (𝐴 ∖ (𝐴 ∩ 𝐶)) ↔ (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝐶)) | 
| 8 | 4, 7 | bitri 275 | 1 ⊢ ((𝐴 ∩ 𝐵) = (𝐴 ∩ 𝐶) ↔ (𝐴 ∖ 𝐵) = (𝐴 ∖ 𝐶)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1539 ∖ cdif 3947 ∩ cin 3949 ⊆ wss 3950 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-3an 1088 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-in 3957 df-ss 3967 | 
| This theorem is referenced by: fressupp 32698 | 
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