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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 4686 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
5 | df-br 4686 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
6 | 3, 4, 5 | 3bitr3g 302 | . 2 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
7 | 1, 2, 6 | eqrelrdv 5250 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1523 ∈ wcel 2030 〈cop 4216 class class class wbr 4685 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-br 4686 df-opab 4746 df-xp 5149 df-rel 5150 |
This theorem is referenced by: funcpropd 16607 fullpropd 16627 fthpropd 16628 dvres 23720 eqfunresadj 31785 |
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