Theorem nfopab 5218

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 1807	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1807	. . 3 ⊢ Ⅎ𝑦⊤
3		nfopab.1	. . . 4 ⊢ Ⅎ𝑧𝜑
4	3	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝜑)
5	1, 2, 4	nfopabd 5217	. 2 ⊢ (⊤ → Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑})
6	5	mptru 1549	1 ⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  ⊤wtru 1543  Ⅎwnf 1786  Ⅎwnfc 2884  {copab 5211

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-opab 5212

This theorem is referenced by:  nfmpt  5256  csbopab  5556  csbopabgALT  5557  nfxp  5710  nfco  5866  nfcnv  5879  nfofr  7677  fineqvrep  34126