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	⊢ 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 1804	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1804	. . 3 ⊢ Ⅎ𝑦⊤
3		nfopab.1	. . . 4 ⊢ Ⅎ𝑧𝜑
4	3	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝜑)
5	1, 2, 4	nfopabd 5211	. 2 ⊢ (⊤ → Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑})
6	5	mptru 1547	1 ⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

Syntax hints:  ⊤wtru 1541  Ⅎwnf 1783  Ⅎwnfc 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