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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 3977 | . . 3 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝐷 ⊆ 𝐵) |
3 | 1 | ineq1i 4173 | . . . 4 ⊢ (𝐴 ∩ 𝐶) = (𝐷 ∩ 𝐶) |
4 | 3 | eqeq1i 2742 | . . 3 ⊢ ((𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶) ↔ (𝐷 ∩ 𝐶) = (𝐵 ∩ 𝐶)) |
5 | 2, 4 | anbi12i 628 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶)) ↔ (𝐷 ⊆ 𝐵 ∧ (𝐷 ∩ 𝐶) = (𝐵 ∩ 𝐶))) |
6 | df-redund 37115 | . 2 ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐴 ⊆ 𝐵 ∧ (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐶))) | |
7 | df-redund 37115 | . 2 ⊢ (𝐷 Redund ⟨𝐵, 𝐶⟩ ↔ (𝐷 ⊆ 𝐵 ∧ (𝐷 ∩ 𝐶) = (𝐵 ∩ 𝐶))) | |
8 | 5, 6, 7 | 3bitr4i 303 | 1 ⊢ (𝐴 Redund ⟨𝐵, 𝐶⟩ ↔ 𝐷 Redund ⟨𝐵, 𝐶⟩) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∩ cin 3914 ⊆ wss 3915 Redund wredund 36684 |
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 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-rab 3411 df-v 3450 df-in 3922 df-ss 3932 df-redund 37115 |
This theorem is referenced by: refrelsredund3 37125 |
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