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| Mirrors > Home > MPE Home > Th. List > eqbrrdiv | Structured version Visualization version GIF version | ||
| Description: Deduction from extensionality principle for relations. (Contributed by Rodolfo Medina, 10-Oct-2010.) | 
| Ref | Expression | 
|---|---|
| eqbrrdiv.1 | ⊢ Rel 𝐴 | 
| eqbrrdiv.2 | ⊢ Rel 𝐵 | 
| eqbrrdiv.3 | ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | 
| Ref | Expression | 
|---|---|
| eqbrrdiv | ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqbrrdiv.1 | . 2 ⊢ Rel 𝐴 | |
| 2 | eqbrrdiv.2 | . 2 ⊢ Rel 𝐵 | |
| 3 | eqbrrdiv.3 | . . 3 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
| 4 | df-br 5144 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
| 5 | df-br 5144 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
| 6 | 3, 4, 5 | 3bitr3g 313 | . 2 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) | 
| 7 | 1, 2, 6 | eqrelrdv 5802 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 〈cop 4632 class class class wbr 5143 Rel wrel 5690 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-v 3482 df-ss 3968 df-br 5144 df-opab 5206 df-xp 5691 df-rel 5692 | 
| This theorem is referenced by: eqfunresadj 7380 funcpropd 17947 fullpropd 17967 fthpropd 17968 dvres 25946 0funcg 48918 0funcALT 48921 functermc2 49141 | 
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