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Theorem dropab1 41071
 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 4787 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
21sps 2186 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ⟨𝑥, 𝑧⟩ = ⟨𝑦, 𝑧⟩)
32eqeq2d 2835 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑤 = ⟨𝑥, 𝑧⟩ ↔ 𝑤 = ⟨𝑦, 𝑧⟩))
43anbi1d 632 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
54drex2 2466 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
65drex1 2465 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑) ↔ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)))
76abbidv 2888 . 2 (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)})
8 df-opab 5115 . 2 {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑥𝑧(𝑤 = ⟨𝑥, 𝑧⟩ ∧ 𝜑)}
9 df-opab 5115 . 2 {⟨𝑦, 𝑧⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑦𝑧(𝑤 = ⟨𝑦, 𝑧⟩ ∧ 𝜑)}
107, 8, 93eqtr4g 2884 1 (∀𝑥 𝑥 = 𝑦 → {⟨𝑥, 𝑧⟩ ∣ 𝜑} = {⟨𝑦, 𝑧⟩ ∣ 𝜑})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536   = wceq 1538  ∃wex 1781  {cab 2802  ⟨cop 4556  {copab 5114 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-13 2392  ax-ext 2796 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 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-v 3482  df-un 3924  df-sn 4551  df-pr 4553  df-op 4557  df-opab 5115 This theorem is referenced by: (None)
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