Theorem nfopab 5174

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	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
nfopab	⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Allowed substitution hints:   𝜑(𝑥, 𝑦, 𝑧)

Proof of Theorem nfopab

Step	Hyp	Ref	Expression
1		nftru 1837	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1837	. . 3 ⊢ Ⅎ𝑦⊤
3		nfopab.1	. . . 4 ⊢ Ⅎ𝑧𝜑
4	3	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝜑)
5	1, 2, 4	nfopabd 5173	. 2 ⊢ (⊤ → Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑})
6	5	mptru 1577	1 ⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

This proof depends on syntax axioms:  ⊤wtru 1571  Ⅎwnf 1816  Ⅎwnfc 2907  {copab 5167

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-opab 5168

This theorem is used by:  nfmpt  5203  csbopab  5534  csbopabw  5535  nfxp  5688  nfco  5845  nfcnv  5858  nfofr  7685  fineqvrep  35640