Theorem nfaba1 2909

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2380. See nfaba1g 2910 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-12 2189. (Revised by SN, 14-May-2025.)

Assertion

Ref	Expression
nfaba1	⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem nfaba1

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-clab 2718	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑} ↔ [𝑧 / 𝑦]∀𝑥𝜑)
2		sbal 2180	. . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
3		nfa1 2162	. . . 4 ⊢ Ⅎ𝑥∀𝑥[𝑧 / 𝑦]𝜑
4	2, 3	nfxfr 1860	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]∀𝑥𝜑
5	1, 4	nfxfr 1860	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑}
6	5	nfci 2889	1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

Syntax hints:  ∀wal 1545  [wsb 2073   ∈ wcel 2119  {cab 2717  Ⅎwnfc 2886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-10 2152  ax-11 2168

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2718  df-nfc 2888

This theorem is referenced by:  nfopd  4821  nfimad  6021  nfiota1  6443  nffvd  6839  nfunidALT2  39461  nfunidALT  39462  nfopdALT  39463  setrec1  50181