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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 5754. (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 5110 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝐴) | |
3 | df-br 5110 | . . . 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 5755 | . 2 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ ((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑)) → 𝐴 = 𝐵) |
6 | 5 | anabss5 667 | 1 ⊢ (((Rel 𝐴 ∧ Rel 𝐵) ∧ 𝜑) → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ⟨cop 4596 class class class wbr 5109 Rel wrel 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3449 df-in 3921 df-ss 3931 df-br 5110 df-opab 5172 df-xp 5643 df-rel 5644 |
This theorem is referenced by: prter3 37394 |
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