Theorem nfaba1 2963

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2379. See nfaba1g 2964 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 2152	. 2 ⊢ Ⅎ𝑥∀𝑥𝜑
2	1	nfab 2961	1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

Syntax hints:  ∀wal 1536  {cab 2776  Ⅎwnfc 2936

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 1911  ax-6 1970  ax-7 2015  ax-10 2142  ax-11 2158  ax-12 2175

This theorem depends on definitions:  df-bi 210  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-nfc 2938

This theorem is referenced by:  nfopd  4782  nfimad  5905  nfiota1  6285  nffvd  6657  nfunidALT2  36265  nfunidALT  36266  nfopdALT  36267  setrec1  45221