Theorem nfopab 5217

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 1801	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1801	. . 3 ⊢ Ⅎ𝑦⊤
3		nfopab.1	. . . 4 ⊢ Ⅎ𝑧𝜑
4	3	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝜑)
5	1, 2, 4	nfopabd 5216	. 2 ⊢ (⊤ → Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑})
6	5	mptru 1544	1 ⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  ⊤wtru 1538  Ⅎwnf 1780  Ⅎwnfc 2888  {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-tru 1540  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:  nfmpt  5255  csbopab  5565  csbopabgALT  5566  nfxp  5722  nfco  5879  nfcnv  5892  nfofr  7704  fineqvrep  35088