Theorem brabidgaw 38497

Description: The law of concretion for a binary relation. Special case of brabga 5480. Version of brabidga 38498 with a disjoint variable condition, which does not require ax-13 2374. (Contributed by Peter Mazsa, 24-Nov-2018.) (Revised by GG, 2-Apr-2024.)

Hypothesis

Ref	Expression
brabidgaw.1	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
brabidgaw	⊢ (𝑥𝑅𝑦 ↔ 𝜑)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝑅(𝑥,𝑦)

Proof of Theorem brabidgaw

Step	Hyp	Ref	Expression
1		brabidgaw.1	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
2	1	breqi 5102	. 2 ⊢ (𝑥𝑅𝑦 ↔ 𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦)
3		df-br 5097	. 2 ⊢ (𝑥{⟨𝑥, 𝑦⟩ ∣ 𝜑}𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑})
4		opabidw 5470	. 2 ⊢ (⟨𝑥, 𝑦⟩ ∈ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ 𝜑)
5	2, 3, 4	3bitri 297	1 ⊢ (𝑥𝑅𝑦 ↔ 𝜑)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  ⟨cop 4584   class class class wbr 5096  {copab 5158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159

This theorem is referenced by:  inxpxrn  38542