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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 4887 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ V) ↔ (〈𝑅, 𝑆〉 ∈ V ∧ 𝑇 ∈ V))) | |
2 | opex 4859 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | biantrur 526 | . 2 ⊢ (𝐶 ∈ V ↔ (〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ V)) |
4 | opex 4859 | . . 3 ⊢ 〈𝑅, 𝑆〉 ∈ V | |
5 | 4 | biantrur 526 | . 2 ⊢ (𝑇 ∈ V ↔ (〈𝑅, 𝑆〉 ∈ V ∧ 𝑇 ∈ V)) |
6 | 1, 3, 5 | 3bitr4g 302 | 1 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → (𝐶 ∈ V ↔ 𝑇 ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 〈cop 4131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 |
This theorem is referenced by: oteqex 4889 |
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