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Theorem nfopabd 5216
Description: Bound-variable hypothesis builder for class abstraction. Deduction form. (Contributed by Scott Fenton, 26-Oct-2024.)
Hypotheses
Ref Expression
nfopabd.1 𝑥𝜑
nfopabd.2 𝑦𝜑
nfopabd.4 (𝜑 → Ⅎ𝑧𝜓)
Assertion
Ref Expression
nfopabd (𝜑𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜓})
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝜓(𝑥,𝑦,𝑧)

Proof of Theorem nfopabd
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 df-opab 5211 . 2 {⟨𝑥, 𝑦⟩ ∣ 𝜓} = {𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)}
2 nfv 1912 . . 3 𝑤𝜑
3 nfopabd.1 . . . 4 𝑥𝜑
4 nfopabd.2 . . . . 5 𝑦𝜑
5 nfvd 1913 . . . . . 6 (𝜑 → Ⅎ𝑧 𝑤 = ⟨𝑥, 𝑦⟩)
6 nfopabd.4 . . . . . 6 (𝜑 → Ⅎ𝑧𝜓)
75, 6nfand 1895 . . . . 5 (𝜑 → Ⅎ𝑧(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
84, 7nfexd 2328 . . . 4 (𝜑 → Ⅎ𝑧𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
93, 8nfexd 2328 . . 3 (𝜑 → Ⅎ𝑧𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓))
102, 9nfabdw 2925 . 2 (𝜑𝑧{𝑤 ∣ ∃𝑥𝑦(𝑤 = ⟨𝑥, 𝑦⟩ ∧ 𝜓)})
111, 10nfcxfrd 2902 1 (𝜑𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜓})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  wex 1776  wnf 1780  {cab 2712  wnfc 2888  cop 4637  {copab 5210
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-opab 5211
This theorem is referenced by:  nfopab  5217  nfttrcld  9748
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