|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > caovord3d | Structured version Visualization version GIF version | ||
| Description: Ordering law. (Contributed by Mario Carneiro, 30-Dec-2014.) | 
| Ref | Expression | 
|---|---|
| caovordg.1 | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | 
| caovordd.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑆) | 
| caovordd.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑆) | 
| caovordd.4 | ⊢ (𝜑 → 𝐶 ∈ 𝑆) | 
| caovord2d.com | ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | 
| caovord3d.5 | ⊢ (𝜑 → 𝐷 ∈ 𝑆) | 
| Ref | Expression | 
|---|---|
| caovord3d | ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | breq1 5145 | . 2 ⊢ ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))) | |
| 2 | caovordg.1 | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆 ∧ 𝑧 ∈ 𝑆)) → (𝑥𝑅𝑦 ↔ (𝑧𝐹𝑥)𝑅(𝑧𝐹𝑦))) | |
| 3 | caovordd.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑆) | |
| 4 | caovordd.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ 𝑆) | |
| 5 | caovordd.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑆) | |
| 6 | caovord2d.com | . . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆)) → (𝑥𝐹𝑦) = (𝑦𝐹𝑥)) | |
| 7 | 2, 3, 4, 5, 6 | caovord2d 7643 | . . 3 ⊢ (𝜑 → (𝐴𝑅𝐶 ↔ (𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵))) | 
| 8 | caovord3d.5 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ 𝑆) | |
| 9 | 2, 8, 5, 4 | caovordd 7642 | . . 3 ⊢ (𝜑 → (𝐷𝑅𝐵 ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵))) | 
| 10 | 7, 9 | bibi12d 345 | . 2 ⊢ (𝜑 → ((𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵) ↔ ((𝐴𝐹𝐵)𝑅(𝐶𝐹𝐵) ↔ (𝐶𝐹𝐷)𝑅(𝐶𝐹𝐵)))) | 
| 11 | 1, 10 | imbitrrid 246 | 1 ⊢ (𝜑 → ((𝐴𝐹𝐵) = (𝐶𝐹𝐷) → (𝐴𝑅𝐶 ↔ 𝐷𝑅𝐵))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1539 ∈ wcel 2107 class class class wbr 5142 (class class class)co 7432 | 
| 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-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-iota 6513 df-fv 6568 df-ov 7435 | 
| This theorem is referenced by: (None) | 
| Copyright terms: Public domain | W3C validator |