Theorem brab 5491

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 5487	. 2 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝐴𝑅𝐵 ↔ 𝜒))
7	1, 2, 6	mp2an 692	1 ⊢ (𝐴𝑅𝐵 ↔ 𝜒)

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3440   class class class wbr 5098  {copab 5160

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 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161

This theorem is referenced by:  opbrop  5722  f1oweALT  7916  frxp  8068  fnwelem  8073  xpord2lem  8084  xpord3lem  8091  poseq  8100  dftpos4  8187  dfac3  10031  axdc2lem  10358  brdom7disj  10441  brdom6disj  10442  ordpipq  10853  ltresr  11051  shftfn  14996  2shfti  15003  ishpg  28831  brcgr  28973  ex-opab  30507  br8d  32686  fineqvnttrclselem3  35279  fineqvnttrclse  35280  vonf1owev  35302  br8  35950  br6  35951  br4  35952  dfbigcup2  36091  brsegle  36302  heiborlem2  38009