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Mirrors > Home > ILE Home > Th. List > otth2 | GIF version |
Description: Ordered triple theorem, with triple express with ordered pairs. (Contributed by NM, 1-May-1995.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
otth.1 | ⊢ 𝐴 ∈ V |
otth.2 | ⊢ 𝐵 ∈ V |
otth.3 | ⊢ 𝑅 ∈ V |
Ref | Expression |
---|---|
otth2 | ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | otth.1 | . . . 4 ⊢ 𝐴 ∈ V | |
2 | otth.2 | . . . 4 ⊢ 𝐵 ∈ V | |
3 | 1, 2 | opth 4237 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)) |
4 | 3 | anbi1i 458 | . 2 ⊢ ((〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆)) |
5 | 1, 2 | opex 4229 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V |
6 | otth.3 | . . 3 ⊢ 𝑅 ∈ V | |
7 | 5, 6 | opth 4237 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 ∧ 𝑅 = 𝑆)) |
8 | df-3an 980 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆) ↔ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ∧ 𝑅 = 𝑆)) | |
9 | 4, 7, 8 | 3bitr4i 212 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝑅〉 = 〈〈𝐶, 𝐷〉, 𝑆〉 ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷 ∧ 𝑅 = 𝑆)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 104 ↔ wb 105 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 Vcvv 2737 〈cop 3595 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 |
This theorem is referenced by: otth 4242 oprabid 5906 eloprabga 5961 |
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