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Theorem dropab1 40198
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 4671 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
21sps 2113 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
32eqeq2d 2782 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑦, 𝑧⟩))
43anbi1d 620 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
54drex2 2378 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
65drex1 2377 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
76abbidv 2837 . 2 (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)})
8 df-opab 4986 . 2 {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)}
9 df-opab 4986 . 2 {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)}
107, 8, 93eqtr4g 2833 1 (∀𝑥 𝑥 = 𝑦 → {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387  wal 1505   = wceq 1507  wex 1742  {cab 2752  cop 4441  {copab 4985
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-rab 3091  df-v 3411  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-opab 4986
This theorem is referenced by: (None)
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