Theorem brabd 38049

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 1943	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax-5 1943	. 2 ⊢ (𝜑 → ∀𝑦𝜑)
3		nfvd 1948	. 2 ⊢ (𝜑 → Ⅎ𝑥𝜒)
4		nfvd 1948	. 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 38048	1 ⊢ (𝜑 → (𝐴𝑅𝐵 ↔ 𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   class class class wbr 5103  {copab 5167

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168

This theorem is used by:  bj-imdirval3  38085