Theorem nfaba1 2934

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2404. See nfaba1g 2935 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-12 2216. (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 2742	. . 3 ⊢ (𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑} ↔ [𝑧 / 𝑦]∀𝑥𝜑)
2		sbal 2209	. . . 4 ⊢ ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
3		nfa1 2191	. . . 4 ⊢ Ⅎ𝑥∀𝑥[𝑧 / 𝑦]𝜑
4	2, 3	nfxfr 1886	. . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]∀𝑥𝜑
5	1, 4	nfxfr 1886	. 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑}
6	5	nfci 2914	1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑}

This proof depends on syntax axioms:  ∀wal 1568  [wsb 2101   ∈ wcel 2147  {cab 2741  Ⅎwnfc 2911

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-10 2180  ax-11 2197

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-nfc 2913

This theorem is used by:  nfopd  4855  nfimad  6083  nfiota1  6522  nffvd  6928  setrec1  10009  nfunidALT2  40292  nfunidALT  40293  nfopdALT  40294