Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  eloprabg GIF version

Theorem eloprabg 5592
 Description: The law of concretion for operation class abstraction. Compare elopab 3995. (Contributed by NM, 14-Sep-1999.) (Revised by David Abernethy, 19-Jun-2012.)
Hypotheses
Ref Expression
eloprabg.1 (𝑥 = 𝐴 → (𝜑𝜓))
eloprabg.2 (𝑦 = 𝐵 → (𝜓𝜒))
eloprabg.3 (𝑧 = 𝐶 → (𝜒𝜃))
Assertion
Ref Expression
eloprabg ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑥,𝐶,𝑦,𝑧   𝜃,𝑥,𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)   𝜒(𝑥,𝑦,𝑧)   𝑉(𝑥,𝑦,𝑧)   𝑊(𝑥,𝑦,𝑧)   𝑋(𝑥,𝑦,𝑧)

Proof of Theorem eloprabg
StepHypRef Expression
1 eloprabg.1 . . 3 (𝑥 = 𝐴 → (𝜑𝜓))
2 eloprabg.2 . . 3 (𝑦 = 𝐵 → (𝜓𝜒))
3 eloprabg.3 . . 3 (𝑧 = 𝐶 → (𝜒𝜃))
41, 2, 3syl3an9b 1205 . 2 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜃))
54eloprabga 5591 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∧ w3a 885   = wceq 1243   ∈ wcel 1393  ⟨cop 3378  {coprab 5513 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-oprab 5516 This theorem is referenced by:  ov  5620  ovg  5639
 Copyright terms: Public domain W3C validator