Theorem brab 4923

Description: The law of concretion for a binary relation. (Contributed by NM, 16-Aug-1999.)

Hypotheses

Ref	Expression
opelopab.1	⊢ 𝐴 ∈ V
opelopab.2	⊢ 𝐵 ∈ V
opelopab.3	⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
opelopab.4	⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
brab.5	⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}

Assertion

Ref	Expression
brab	⊢ (𝐴𝑅𝐵 ↔ 𝜒)

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

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

Proof of Theorem brab

Step	Hyp	Ref	Expression
1		opelopab.1	. 2 ⊢ 𝐴 ∈ V
2		opelopab.2	. 2 ⊢ 𝐵 ∈ V
3		opelopab.3	. . 3 ⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜓))
4		opelopab.4	. . 3 ⊢ (𝑦 = 𝐵 → (𝜓 ↔ 𝜒))
5		brab.5	. . 3 ⊢ 𝑅 = {⟨𝑥, 𝑦⟩ ∣ 𝜑}
6	3, 4, 5	brabg 4919	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒))
7	1, 2, 6	mp2an 704	1 ⊢ (𝐴𝑅𝐵 ↔ 𝜒)

Syntax hints:   → wi 4   ↔ wb 195   = wceq 1475   ∈ wcel 1977  Vcvv 3173   class class class wbr 4583  {copab 4642

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644

This theorem is referenced by:  opbrop  5121  f1oweALT  7043  frxp  7174  fnwelem  7179  dftpos4  7258  dfac3  8827  axdc2lem  9153  brdom7disj  9234  brdom6disj  9235  ordpipq  9643  ltresr  9840  shftfn  13661  2shfti  13668  ishpg  25451  brcgr  25580  ex-opab  26681  br8d  28802  br8  30899  br6  30900  br4  30901  poseq  30994  dfbigcup2  31176  brsegle  31385  heiborlem2  32781