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Mirrors > Home > MPE Home > Th. List > oteqex2 | Structured version Visualization version GIF version |
Description: Equivalence of existence implied by equality of ordered triples. (Contributed by NM, 26-Apr-2015.) |
Ref | Expression |
---|---|
oteqex2 | ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → (𝐶 ∈ V ↔ 𝑇 ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeqex 5387 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ V) ↔ (〈𝑅, 𝑆〉 ∈ V ∧ 𝑇 ∈ V))) | |
2 | opex 5355 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | biantrur 533 | . 2 ⊢ (𝐶 ∈ V ↔ (〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ V)) |
4 | opex 5355 | . . 3 ⊢ 〈𝑅, 𝑆〉 ∈ V | |
5 | 4 | biantrur 533 | . 2 ⊢ (𝑇 ∈ V ↔ (〈𝑅, 𝑆〉 ∈ V ∧ 𝑇 ∈ V)) |
6 | 1, 3, 5 | 3bitr4g 316 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → (𝐶 ∈ V ↔ 𝑇 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 Vcvv 3494 〈cop 4572 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 |
This theorem is referenced by: oteqex 5389 |
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