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Theorem dropab2 42632
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
dropab2 (∀𝑥 𝑥 = 𝑦 → {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜑})

Proof of Theorem dropab2
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 opeq2 4829 . . . . . . . 8 (𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
21sps 2178 . . . . . . 7 (∀𝑥 𝑥 = 𝑦 → ⟨𝑧, 𝑥⟩ = ⟨𝑧, 𝑦⟩)
32eqeq2d 2748 . . . . . 6 (∀𝑥 𝑥 = 𝑦 → (𝑤 = ⟨𝑧, 𝑥⟩ ↔ 𝑤 = ⟨𝑧, 𝑦⟩))
43anbi1d 630 . . . . 5 (∀𝑥 𝑥 = 𝑦 → ((𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ (𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
54drex1 2440 . . . 4 (∀𝑥 𝑥 = 𝑦 → (∃𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
65drex2 2441 . . 3 (∀𝑥 𝑥 = 𝑦 → (∃𝑧𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑) ↔ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)))
76abbidv 2806 . 2 (∀𝑥 𝑥 = 𝑦 → {𝑤 ∣ ∃𝑧𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)})
8 df-opab 5166 . 2 {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧𝑥(𝑤 = ⟨𝑧, 𝑥⟩ ∧ 𝜑)}
9 df-opab 5166 . 2 {⟨𝑧, 𝑦⟩ ∣ 𝜑} = {𝑤 ∣ ∃𝑧𝑦(𝑤 = ⟨𝑧, 𝑦⟩ ∧ 𝜑)}
107, 8, 93eqtr4g 2802 1 (∀𝑥 𝑥 = 𝑦 → {⟨𝑧, 𝑥⟩ ∣ 𝜑} = {⟨𝑧, 𝑦⟩ ∣ 𝜑})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wal 1539   = wceq 1541  wex 1781  {cab 2714  cop 4590  {copab 5165
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-13 2371  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-opab 5166
This theorem is referenced by: (None)
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