Theorem brabidga 38322

Description: The law of concretion for a binary relation. Special case of brabga 5553. Usage of this theorem is discouraged because it depends on ax-13 2380, see brabidgaw 38321 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	⊢ (𝑥𝑅𝑦 ↔ 𝜑)

Proof of Theorem brabidga

Step	Hyp	Ref	Expression
1		brabidga.1	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	breqi 5172	. 2 ⊢ (𝑥𝑅𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
3		df-br 5167	. 2 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4		opabid 5544	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
5	2, 3, 4	3bitri 297	1 ⊢ (𝑥𝑅𝑦 ↔ 𝜑)

Syntax hints:   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ⟨cop 4654   class class class wbr 5166  {copab 5228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2178  ax-13 2380  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229

This theorem is referenced by: (None)