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Theorem brrabga 35751
Description: The law of concretion for operation class abstraction. (Contributed by Peter Mazsa, 24-Oct-2022.)
Hypotheses
Ref Expression
brrabga.1 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
brrabga.2 𝑅 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
Assertion
Ref Expression
brrabga ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵𝑅𝐶𝜓))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜓,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝑅(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem brrabga
StepHypRef Expression
1 df-br 5034 . . 3 (⟨𝐴, 𝐵𝑅𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑅)
2 brrabga.2 . . . 4 𝑅 = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑}
32eleq2i 2884 . . 3 (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ 𝑅 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
41, 3bitri 278 . 2 (⟨𝐴, 𝐵𝑅𝐶 ↔ ⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑})
5 brrabga.1 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜓))
65eloprabga 7244 . 2 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜓))
74, 6syl5bb 286 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨𝐴, 𝐵𝑅𝐶𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  w3a 1084   = wceq 1538  wcel 2112  cop 4534   class class class wbr 5033  {coprab 7140
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-oprab 7143
This theorem is referenced by:  brcnvrabga  35752
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