Theorem brabidga 38341

Description: The law of concretion for a binary relation. Special case of brabga 5489. Usage of this theorem is discouraged because it depends on ax-13 2370, see brabidgaw 38340 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 5108	. 2 ⊢ (𝑥𝑅𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
3		df-br 5103	. 2 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4		opabid 5480	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
5	2, 3, 4	3bitri 297	1 ⊢ (𝑥𝑅𝑦 ↔ 𝜑)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ⟨cop 4591   class class class wbr 5102  {copab 5164

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-13 2370  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165

This theorem is referenced by: (None)