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 5660. (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 5058 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | df-br 5058 | . . . 4 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
4 | 1, 2, 3 | 3bitr3g 314 | . . 3 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
5 | 4 | eqrelrdv2 5661 | . 2 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ ((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑)) → 𝐴 = 𝐵) |
6 | 5 | anabss5 664 | 1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 〈cop 4563 class class class wbr 5057 Rel wrel 5553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-in 3940 df-ss 3949 df-br 5058 df-opab 5120 df-xp 5554 df-rel 5555 |
This theorem is referenced by: prter3 35898 |
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