Theorem nfaba1 2900

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 2902 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-6 1967, ax-7 2008, ax-12 2178. (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 2709	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑} ↔ [𝑧 / 𝑦]∀𝑥𝜑)
2		sbal 2170	. . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
3		nfa1 2152	. . . 4 ⊢ Ⅎ𝑥∀𝑥[𝑧 / 𝑦]𝜑
4	2, 3	nfxfr 1853	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]∀𝑥𝜑
5	1, 4	nfxfr 1853	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑}
6	5	nfci 2880	1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

Syntax hints:  ∀wal 1538  [wsb 2065   ∈ wcel 2109  {cab 2708  Ⅎwnfc 2877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-10 2142  ax-11 2158

This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2709  df-nfc 2879

This theorem is referenced by:  nfopd  4857  nfimad  6043  nfiota1  6469  nffvd  6873  nfunidALT2  38969  nfunidALT  38970  nfopdALT  38971  setrec1  49684