Theorem nfaba1 2988

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2390. See nfaba1g 2989 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 2155	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nfab 2986	1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

Syntax hints:  ∀wal 1535  {cab 2801  Ⅎwnfc 2963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-nfc 2965

This theorem is referenced by:  nfopd  4822  nfimad  5940  nfiota1  6318  nffvd  6684  nfunidALT2  36107  nfunidALT  36108  nfopdALT  36109  setrec1  44801