Theorem braba 4022

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 4019	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜓))
6	1, 2, 5	mp2an 416	1 ⊢ (𝐴𝑅𝐵 ↔ 𝜓)

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284   ∈ wcel 1433  Vcvv 2601   class class class wbr 3785  {copab 3838

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840

This theorem is referenced by: (None)