Theorem braba 5475

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 5472	. 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 1541   ∈ wcel 2111  Vcvv 3436   class class class wbr 5089  {copab 5151

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 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

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-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152

This theorem is referenced by:  frgpuplem  19684  2ndcctbss  23370  legov  28563  prtlem13  38915  wepwsolem  43083  fnwe2val  43090  grumnud  44327  lambert0  46926  lamberte  46927  sinnpoly  46930  sprsymrelf  47534  catprsc  49053  catprsc2  49054