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 2366. See nfaba1g 2902 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-6 1964, ax-7 2004, ax-12 2167. (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 2704	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑} ↔ [𝑧 / 𝑦]∀𝑥𝜑)
2		sbal 2159	. . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
3		nfa1 2141	. . . 4 ⊢ Ⅎ𝑥∀𝑥[𝑧 / 𝑦]𝜑
4	2, 3	nfxfr 1848	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]∀𝑥𝜑
5	1, 4	nfxfr 1848	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑}
6	5	nfci 2879	1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

Syntax hints:  ∀wal 1532  [wsb 2060   ∈ wcel 2099  {cab 2703  Ⅎwnfc 2876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-10 2130  ax-11 2147

This theorem depends on definitions:  df-bi 206  df-or 846  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-nfc 2878

This theorem is referenced by:  nfopd  4896  nfimad  6078  nfiota1  6508  nffvd  6913  nfunidALT2  38667  nfunidALT  38668  nfopdALT  38669  setrec1  48437