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Theorem eloprabg 5906
 Description: The law of concretion for operation class abstraction. Compare elopab 4218. (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 1292 . 2 ((𝑥 = 𝐴𝑦 = 𝐵𝑧 = 𝐶) → (𝜑𝜃))
54eloprabga 5905 1 ((𝐴𝑉𝐵𝑊𝐶𝑋) → (⟨⟨𝐴, 𝐵⟩, 𝐶⟩ ∈ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝜑} ↔ 𝜃))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104   ∧ w3a 963   = wceq 1335   ∈ wcel 2128  ⟨cop 3563  {coprab 5822 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-oprab 5825 This theorem is referenced by:  ov  5937  ovg  5956
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