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Mirrors > Home > MPE Home > Th. List > Mathboxes > redundeq1 | Structured version Visualization version GIF version |
Description: Equivalence of redundancy predicates. (Contributed by Peter Mazsa, 26-Oct-2022.) |
Ref | Expression |
---|---|
redundeq1.1 | ⊢ 𝐴 = 𝐷 |
Ref | Expression |
---|---|
redundeq1 | ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 ↔ 𝐷 Redund 〈𝐵, 𝐶〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | redundeq1.1 | . . . 4 ⊢ 𝐴 = 𝐷 | |
2 | 1 | sseq1i 3943 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐷 ⊆ 𝐵) |
3 | 1 | ineq1i 4135 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐷 ∩ 𝐶) |
4 | 3 | eqeq1i 2803 | . . 3 ⊢ ((𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) ↔ (𝐷 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
5 | 2, 4 | anbi12i 629 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) ↔ (𝐷 ⊆ 𝐵 ∧ (𝐷 ∩ 𝐶) = (𝐵 ∩ 𝐶))) |
6 | df-redund 36019 | . 2 ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))) | |
7 | df-redund 36019 | . 2 ⊢ (𝐷 Redund 〈𝐵, 𝐶〉 ↔ (𝐷 ⊆ 𝐵 ∧ (𝐷 ∩ 𝐶) = (𝐵 ∩ 𝐶))) | |
8 | 5, 6, 7 | 3bitr4i 306 | 1 ⊢ (𝐴 Redund 〈𝐵, 𝐶〉 ↔ 𝐷 Redund 〈𝐵, 𝐶〉) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∧ wa 399 = wceq 1538 ∩ cin 3880 ⊆ wss 3881 Redund wredund 35634 |
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 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-v 3443 df-in 3888 df-ss 3898 df-redund 36019 |
This theorem is referenced by: refrelsredund3 36029 |
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