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Theorem brabsb2 37374
Description: A closed form of brabsb 5492. (Contributed by Rodolfo Medina, 13-Oct-2010.)
Assertion
Ref Expression
brabsb2 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Distinct variable groups:   𝑥,𝑦,𝑧   𝑥,𝑤,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑤)   𝑅(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem brabsb2
StepHypRef Expression
1 breq 5111 . . 3 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑤))
2 df-br 5110 . . 3 (𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
31, 2bitrdi 287 . 2 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ ⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑}))
4 vopelopabsb 5490 . 2 (⟨𝑧, 𝑤⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
53, 4bitrdi 287 1 (𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑} → (𝑧𝑅𝑤 ↔ [𝑧 / 𝑥][𝑤 / 𝑦]𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  [wsb 2068  wcel 2107  cop 4596   class class class wbr 5109  {copab 5171
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-10 2138  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172
This theorem is referenced by: (None)
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