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| Mirrors > Home > MPE Home > Th. List > Mathboxes > brabsb2 | Structured version Visualization version GIF version | ||
| Description: A closed form of brabsb 5535. (Contributed by Rodolfo Medina, 13-Oct-2010.) | 
| Ref | Expression | 
|---|---|
| brabsb2 | ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | breq 5144 | . . 3 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑤)) | |
| 2 | df-br 5143 | . . 3 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑤 ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
| 3 | 1, 2 | bitrdi 287 | . 2 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) | 
| 4 | vopelopabsb 5533 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | |
| 5 | 3, 4 | bitrdi 287 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 [wsb 2063 ∈ wcel 2107 〈cop 4631 class class class wbr 5142 {copab 5204 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 | 
| This theorem is referenced by: (None) | 
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