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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 5391 | . 2 ⊢ (〈〈𝐴, 𝐵〉, 𝐶〉 = 〈〈𝑅, 𝑆〉, 𝑇〉 → ((〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ V) ↔ (〈𝑅, 𝑆〉 ∈ V ∧ 𝑇 ∈ V))) | |
2 | opex 5359 | . . 3 ⊢ 〈𝐴, 𝐵〉 ∈ V | |
3 | 2 | biantrur 533 | . 2 ⊢ (𝐶 ∈ V ↔ (〈𝐴, 𝐵〉 ∈ V ∧ 𝐶 ∈ V)) |
4 | opex 5359 | . . 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 1536 ∈ wcel 2113 Vcvv 3497 〈cop 4576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-v 3499 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 |
This theorem is referenced by: oteqex 5393 |
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