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Theorem nfaba1 2911
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2375. See nfaba1g 2913 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-6 1965, ax-7 2005, ax-12 2175. (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 2713 . . 3 (𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑} ↔ [𝑧 / 𝑦]∀𝑥𝜑)
2 sbal 2167 . . . 4 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
3 nfa1 2149 . . . 4 𝑥𝑥[𝑧 / 𝑦]𝜑
42, 3nfxfr 1850 . . 3 𝑥[𝑧 / 𝑦]∀𝑥𝜑
51, 4nfxfr 1850 . 2 𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑}
65nfci 2891 1 𝑥{𝑦 ∣ ∀𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wal 1535  [wsb 2062  wcel 2106  {cab 2712  wnfc 2888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-10 2139  ax-11 2155
This theorem depends on definitions:  df-bi 207  df-or 848  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-nfc 2890
This theorem is referenced by:  nfopd  4895  nfimad  6089  nfiota1  6518  nffvd  6919  nfunidALT2  38951  nfunidALT  38952  nfopdALT  38953  setrec1  48922
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