Theorem nfaba1 2916

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 2918 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-6 1967, ax-7 2007, 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 2718	. . 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 1851	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]∀𝑥𝜑
5	1, 4	nfxfr 1851	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑}
6	5	nfci 2896	1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

Syntax hints:  ∀wal 1535  [wsb 2064   ∈ wcel 2108  {cab 2717  Ⅎwnfc 2893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-10 2141  ax-11 2158

This theorem depends on definitions:  df-bi 207  df-or 847  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-nfc 2895

This theorem is referenced by:  nfopd  4914  nfimad  6098  nfiota1  6527  nffvd  6932  nfunidALT2  38925  nfunidALT  38926  nfopdALT  38927  setrec1  48783