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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 2253 | . . 3 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (𝑥 ∈ 〈𝐴, 𝐵〉 ↔ 𝑥 ∈ 〈𝐶, 𝐷〉)) | |
2 | 1 | exbidv 1836 | . 2 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 ↔ ∃𝑥 𝑥 ∈ 〈𝐶, 𝐷〉)) |
3 | opm 4255 | . 2 ⊢ (∃𝑥 𝑥 ∈ 〈𝐴, 𝐵〉 ↔ (𝐴 ∈ V ∧ 𝐵 ∈ V)) | |
4 | opm 4255 | . 2 ⊢ (∃𝑥 𝑥 ∈ 〈𝐶, 𝐷〉 ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V)) | |
5 | 2, 3, 4 | 3bitr3g 222 | 1 ⊢ (〈𝐴, 𝐵〉 = 〈𝐶, 𝐷〉 → ((𝐴 ∈ V ∧ 𝐵 ∈ V) ↔ (𝐶 ∈ V ∧ 𝐷 ∈ V))) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∃wex 1503 ∈ wcel 2160 Vcvv 2752 〈cop 3613 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4139 ax-pow 4195 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 |
This theorem is referenced by: epelg 4311 |
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