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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 4001 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐷 ⊆ 𝐵) |
3 | 1 | ineq1i 4202 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐷 ∩ 𝐶) |
4 | 3 | eqeq1i 2730 | . . 3 ⊢ ((𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) ↔ (𝐷 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
5 | 2, 4 | anbi12i 626 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) ↔ (𝐷 ⊆ 𝐵 ∧ (𝐷 ∩ 𝐶) = (𝐵 ∩ 𝐶))) |
6 | df-redund 38152 | . 2 ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))) | |
7 | df-redund 38152 | . 2 ⊢ (𝐷 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐷 ⊆ 𝐵 ∧ (𝐷 ∩ 𝐶) = (𝐵 ∩ 𝐶))) | |
8 | 5, 6, 7 | 3bitr4i 302 | 1 ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ 𝐷 Redund ⟨𝐵, 𝐶⟩) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 394 = wceq 1533 ∩ cin 3938 ⊆ wss 3939 Redund wredund 37726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-sb 2060 df-clab 2703 df-cleq 2717 df-clel 2802 df-rab 3420 df-in 3946 df-ss 3956 df-redund 38152 |
This theorem is referenced by: refrelsredund3 38162 |
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