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Mathbox for Rodolfo Medina |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqbrrdv2 | Structured version Visualization version GIF version |
Description: Other version of eqbrrdiv 5807. (Contributed by Rodolfo Medina, 30-Sep-2010.) |
Ref | Expression |
---|---|
eqbrrdv2.1 | ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
Ref | Expression |
---|---|
eqbrrdv2 | ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrrdv2.1 | . . . 4 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
2 | df-br 5149 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | df-br 5149 | . . . 4 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
4 | 1, 2, 3 | 3bitr3g 313 | . . 3 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
5 | 4 | eqrelrdv2 5808 | . 2 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ ((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑)) → 𝐴 = 𝐵) |
6 | 5 | anabss5 668 | 1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 〈cop 4637 class class class wbr 5148 Rel wrel 5694 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-ss 3980 df-br 5149 df-opab 5211 df-xp 5695 df-rel 5696 |
This theorem is referenced by: prter3 38864 |
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