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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 𝑥{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.
StepHypRef 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 𝑥𝑥[𝑧 / 𝑦]𝜑
42, 3nfxfr 1848 . . 3 𝑥[𝑧 / 𝑦]∀𝑥𝜑
51, 4nfxfr 1848 . 2 𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑}
65nfci 2879 1 𝑥{𝑦 ∣ ∀𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wal 1532  [wsb 2060  wcel 2099  {cab 2703  wnfc 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
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