Theorem brabd 37508

Description: Expressing that two sets are related by a binary relation which is expressed as a class abstraction of ordered pairs. (Contributed by BJ, 17-Dec-2023.)

Hypotheses

Ref	Expression
brabd.exa	⊢ (𝜑 → 𝐴 ∈ 𝑈)
brabd.exb	⊢ (𝜑 → 𝐵 ∈ 𝑉)
brabd.def	⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
brabd.is	⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
brabd	⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))

Distinct variable groups:   𝑥,𝐴,𝑦   𝑥,𝐵,𝑦   𝜑,𝑥,𝑦   𝜒,𝑥,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝑅(𝑥,𝑦)   𝑈(𝑥,𝑦)   𝑉(𝑥,𝑦)

Proof of Theorem brabd

Step	Hyp	Ref	Expression
1		ax-5 1917	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax-5 1917	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		nfvd 1922	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4		nfvd 1922	. 2 ⊢ (𝜑 → Ⅎ𝑦𝜒)
5		brabd.exa	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈)
6		brabd.exb	. 2 ⊢ (𝜑 → 𝐵 ∈ 𝑉)
7		brabd.def	. 2 ⊢ (𝜑 → 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜓})
8		brabd.is	. 2 ⊢ ((𝜑 ∧ (𝑥 = 𝐴 ∧ 𝑦 = 𝐵)) → (𝜓 ↔ 𝜒))
9	1, 2, 3, 4, 5, 6, 7, 8	brabd0 37507	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119   class class class wbr 5072  {copab 5134

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135

This theorem is referenced by:  bj-imdirval3  37544