Theorem nfaba1 2911

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2371. See nfaba1g 2912 for a less restrictive version requiring more axioms. (Revised by Gino Giotto, 20-Jan-2024.)

Assertion

Ref	Expression
nfaba1	⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem nfaba1

Step	Hyp	Ref	Expression
1		nfa1 2148	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nfab 2909	1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

Syntax hints:  ∀wal 1539  {cab 2709  Ⅎwnfc 2883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-10 2137  ax-11 2154  ax-12 2171

This theorem depends on definitions:  df-bi 206  df-or 846  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-nfc 2885

This theorem is referenced by:  nfopd  4890  nfimad  6068  nfiota1  6497  nffvd  6903  nfunidALT2  37834  nfunidALT  37835  nfopdALT  37836  setrec1  47726