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Theorem brabidga 35605
 Description: The law of concretion for a binary relation. Special case of brabga 5412. Usage of this theorem is discouraged because it depends on ax-13 2383, see brabidgaw 35604 for a weaker version that does not require it. (Contributed by Peter Mazsa, 24-Nov-2018.) (New usage is discouraged.)
Hypothesis
Ref Expression
brabidga.1 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
Assertion
Ref Expression
brabidga (𝑥𝑅𝑦𝜑)

Proof of Theorem brabidga
StepHypRef Expression
1 brabidga.1 . . 3 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
21breqi 5063 . 2 (𝑥𝑅𝑦𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
3 df-br 5058 . 2 (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4 opabid 5404 . 2 (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
52, 3, 43bitri 299 1 (𝑥𝑅𝑦𝜑)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 208   = wceq 1530   ∈ wcel 2107  ⟨cop 4565   class class class wbr 5057  {copab 5119 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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2383  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120 This theorem is referenced by: (None)
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