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Mirrors > Home > MPE Home > Th. List > eqrelrdv2 | Structured version Visualization version GIF version |
Description: A version of eqrelrdv 5250. (Contributed by Rodolfo Medina, 10-Oct-2010.) |
Ref | Expression |
---|---|
eqrelrdv2.1 | ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
Ref | Expression |
---|---|
eqrelrdv2 | ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqrelrdv2.1 | . . . 4 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
2 | 1 | alrimivv 1896 | . . 3 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
3 | eqrel 5243 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
4 | 2, 3 | syl5ibr 236 | . 2 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝜑 → 𝐴 = 𝐵)) |
5 | 4 | imp 444 | 1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∀wal 1521 = wceq 1523 ∈ wcel 2030 〈cop 4216 Rel wrel 5148 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-in 3614 df-ss 3621 df-opab 4746 df-xp 5149 df-rel 5150 |
This theorem is referenced by: xpiindi 5290 fliftcnv 6601 dmtpos 7409 ercnv 7808 fpwwe2lem9 9498 invsym2 16470 eqbrrdv2 34467 dibglbN 36772 diclspsn 36800 dih1 36892 dihglbcpreN 36906 dihmeetlem4preN 36912 rfovcnvf1od 38615 |
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