Users' Mathboxes Mathbox for Andrew Salmon < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dropab1 Structured version   Visualization version   GIF version

Theorem dropab1 38477
Description: Theorem to aid use of the distinctor reduction theorem with ordered pair class abstraction. (Contributed by Andrew Salmon, 25-Jul-2011.)
Assertion
Ref Expression
dropab1 (∀𝑥 𝑥 = 𝑦 → {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ 𝜑})

Proof of Theorem dropab1
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 opeq1 4400 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
21sps 2054 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
32eqeq2d 2631 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑦, 𝑧⟩))
43anbi1d 741 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
54drex2 2327 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
65drex1 2326 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
76abbidv 2740 . 2 (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)})
8 df-opab 4711 . 2 {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)}
9 df-opab 4711 . 2 {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)}
107, 8, 93eqtr4g 2680 1 (∀𝑥 𝑥 = 𝑦 → {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wal 1480   = wceq 1482  wex 1703  {cab 2607  cop 4181  {copab 4710
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-opab 4711
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator