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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 5106 | . . 3 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
5 | df-br 5106 | . . 3 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
6 | 3, 4, 5 | 3bitr3g 312 | . 2 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
7 | 1, 2, 6 | eqrelrdv 5748 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ∈ wcel 2106 〈cop 4592 class class class wbr 5105 Rel wrel 5638 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-v 3447 df-in 3917 df-ss 3927 df-br 5106 df-opab 5168 df-xp 5639 df-rel 5640 |
This theorem is referenced by: eqfunresadj 7304 funcpropd 17786 fullpropd 17806 fthpropd 17807 dvres 25273 |
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