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| Description: Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.) | 
| Ref | Expression | 
|---|---|
| eqrelrdv.1 | ⊢ Rel 𝐴 | 
| eqrelrdv.2 | ⊢ Rel 𝐵 | 
| eqrelrdv.3 | ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) | 
| Ref | Expression | 
|---|---|
| eqrelrdv | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqrelrdv.3 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) | |
| 2 | 1 | alrimivv 1927 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) | 
| 3 | eqrelrdv.1 | . . 3 ⊢ Rel 𝐴 | |
| 4 | eqrelrdv.2 | . . 3 ⊢ Rel 𝐵 | |
| 5 | eqrel 5793 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
| 6 | 3, 4, 5 | mp2an 692 | . 2 ⊢ (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) | 
| 7 | 2, 6 | sylibr 234 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∀wal 1537 = wceq 1539 ∈ wcel 2107 〈cop 4631 Rel wrel 5689 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-ss 3967 df-opab 5205 df-xp 5690 df-rel 5691 | 
| This theorem is referenced by: eqbrrdiv 5803 fcnvres 6784 fmptco 7148 fpwwe2lem7 10678 fpwwe2lem11 10682 fsumcom2 15811 fprodcom2 16021 gsumcom2 19994 lgsquadlem1 27425 lgsquadlem2 27426 fmptcof2 32668 dfcnv2 32687 dih1dimatlem 41332 | 
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