Theorem nfopab 5173

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 1836	. . 3 ⊢ Ⅎ𝑥⊤
2		nftru 1836	. . 3 ⊢ Ⅎ𝑦⊤
3		nfopab.1	. . . 4 ⊢ Ⅎ𝑧𝜑
4	3	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑧𝜑)
5	1, 2, 4	nfopabd 5172	. 2 ⊢ (⊤ → Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑})
6	5	mptru 1576	1 ⊢ Ⅎ𝑧{⟨𝑥, 𝑦⟩ ∣ 𝜑}

This proof depends on syntax axioms:  ⊤wtru 1570  Ⅎwnf 1815  Ⅎwnfc 2907  {copab 5166

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-opab 5167

This theorem is used by:  nfmpt  5202  csbopab  5529  csbopabw  5530  nfxp  5683  nfco  5843  nfcnv  5856  nfofr  7702  fineqvrep  35840  srepdflem1  36006  srepdflem2  36007