Theorem braba 5500

Description: The law of concretion for a binary relation. (Contributed by NM, 19-Dec-2013.)

Hypotheses

Ref	Expression
opelopaba.1	⊢ 𝐴 ∈ V
opelopaba.2	⊢ 𝐵 ∈ V
opelopaba.3	⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
braba.4	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
braba	⊢ (𝐴𝑅𝐵 ↔ 𝜓)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦

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

Proof of Theorem braba

Step	Hyp	Ref	Expression
1		opelopaba.1	. 2 ⊢ 𝐴 ∈ V
2		opelopaba.2	. 2 ⊢ 𝐵 ∈ V
3		opelopaba.3	. . 3 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝜑 ↔ 𝜓))
4		braba.4	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
5	3, 4	brabga 5497	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜓))
6	1, 2, 5	mp2an 692	1 ⊢ (𝐴𝑅𝐵 ↔ 𝜓)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  Vcvv 3450   class class class wbr 5110  {copab 5172

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-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173

This theorem is referenced by:  frgpuplem  19709  2ndcctbss  23349  legov  28519  prtlem13  38868  wepwsolem  43038  fnwe2val  43045  grumnud  44282  lambert0  46895  lamberte  46896  sprsymrelf  47500  catprsc  49006  catprsc2  49007