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Theorem nfopab 5212
Description: Bound-variable hypothesis builder for class abstraction. (Contributed by NM, 1-Sep-1999.) Remove disjoint variable conditions. (Revised by Andrew Salmon, 11-Jul-2011.) (Revised by Scott Fenton, 26-Oct-2024.)
Hypothesis
Ref Expression
nfopab.1 𝑧𝜑
Assertion
Ref Expression
nfopab 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Distinct variable groups:   𝑥,𝑧   𝑦,𝑧
Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)

Proof of Theorem nfopab
StepHypRef Expression
1 nftru 1804 . . 3 𝑥
2 nftru 1804 . . 3 𝑦
3 nfopab.1 . . . 4 𝑧𝜑
43a1i 11 . . 3 (⊤ → Ⅎ𝑧𝜑)
51, 2, 4nfopabd 5211 . 2 (⊤ → 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑})
65mptru 1547 1 𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}
Colors of variables: wff setvar class
Syntax hints:  wtru 1541  wnf 1783  wnfc 2890  {copab 5205
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-opab 5206
This theorem is referenced by:  nfmpt  5249  csbopab  5560  csbopabgALT  5561  nfxp  5718  nfco  5876  nfcnv  5889  nfofr  7704  fineqvrep  35109
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