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Mirrors > Home > MPE Home > Th. List > Mathboxes > brabsb2 | Structured version Visualization version GIF version |
Description: A closed form of brabsb 5426. (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 5069 | . . 3 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑤)) | |
2 | df-br 5068 | . . 3 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑤 ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
3 | 1, 2 | bitrdi 290 | . 2 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) |
4 | vopelopabsb 5424 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑) | |
5 | 3, 4 | bitrdi 290 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 [wsb 2071 ∈ wcel 2111 〈cop 4561 class class class wbr 5067 {copab 5129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-br 5068 df-opab 5130 |
This theorem is referenced by: (None) |
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