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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 2204 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (𝑥 ∈ 〈𝐴, 𝐵〉 ↔ 𝑥 ∈ 〈𝐶, 𝐷〉)) | |
2 | 1 | exbidv 1798 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 ↔ ∃𝑥 𝑥 ∈ 〈𝐶, 𝐷〉)) |
3 | opm 4164 | . 2 ⊢ (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | opm 4164 | . 2 ⊢ (∃𝑥 𝑥 ∈ 〈𝐶, 𝐷〉 ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)) | |
5 | 2, 3, 4 | 3bitr3g 221 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1332 ∃wex 1469 ∈ wcel 1481 Vcvv 2689 〈cop 3535 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 |
This theorem is referenced by: epelg 4220 |
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