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Mirrors > Home > ILE Home > Th. List > opeqex | GIF version |
Description: Equivalence of existence implied by equality of ordered pairs. (Contributed by NM, 28-May-2008.) |
Ref | Expression |
---|---|
opeqex | ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2142 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (𝑥 ∈ 〈𝐴, 𝐵〉 ↔ 𝑥 ∈ 〈𝐶, 𝐷〉)) | |
2 | 1 | exbidv 1746 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 ↔ ∃𝑥 𝑥 ∈ 〈𝐶, 𝐷〉)) |
3 | opm 3989 | . 2 ⊢ (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | opm 3989 | . 2 ⊢ (∃𝑥 𝑥 ∈ 〈𝐶, 𝐷〉 ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)) | |
5 | 2, 3, 4 | 3bitr3g 220 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ↔ wb 103 = wceq 1284 ∃wex 1421 ∈ wcel 1433 Vcvv 2601 〈cop 3401 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-v 2603 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 |
This theorem is referenced by: epelg 4045 |
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