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Mathbox for Rodolfo Medina |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > brabsb2 | Structured version Visualization version GIF version |
Description: A closed form of brabsb 4911. (Contributed by Rodolfo Medina, 13-Oct-2010.) |
Ref | Expression |
---|---|
brabsb2 | ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq 4585 | . . 3 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ 𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑤)) | |
2 | df-br 4584 | . . 3 ⊢ (𝑧{〈𝑥, 𝑦〉 ∣ 𝜑}𝑤 ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑}) | |
3 | 1, 2 | syl6bb 275 | . 2 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ 〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑})) |
4 | opelopabsbALT 4909 | . 2 ⊢ (〈𝑧, 𝑤〉 ∈ {〈𝑥, 𝑦〉 ∣ 𝜑} ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑) | |
5 | 3, 4 | syl6bb 275 | 1 ⊢ (𝑅 = {〈𝑥, 𝑦〉 ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑤 / 𝑦][𝑧 / 𝑥]𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 = wceq 1475 [wsb 1867 ∈ wcel 1977 〈cop 4131 class class class wbr 4583 {copab 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 |
This theorem is referenced by: (None) |
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