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Theorem brabsb 5522
Description: The law of concretion in terms of substitutions. (Contributed by NM, 17-Mar-2008.)
Hypothesis
Ref Expression
brabsb.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brabsb (𝐴𝑅𝐵[𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
Distinct variable groups:   𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)   𝐵(𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brabsb
StepHypRef Expression
1 df-br 5140 . 2 (𝐴𝑅𝐵 ↔ ⟨𝐴, 𝐵⟩ ∈ 𝑅)
2 brabsb.1 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
32eleq2i 2817 . 2 (⟨𝐴, 𝐵⟩ ∈ 𝑅 ↔ ⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 opelopabsb 5521 . 2 (⟨𝐴, 𝐵⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ [𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
51, 3, 43bitri 297 1 (𝐴𝑅𝐵[𝐴 / 𝑥][𝐵 / 𝑦]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  [wsbc 3770  cop 4627   class class class wbr 5139  {copab 5201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3771  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202
This theorem is referenced by:  eqerlem  8734  xpab  35219
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