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Theorem nfaba1 2913
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 14-Oct-2016.) Add disjoint variable condition to avoid ax-13 2377. See nfaba1g 2915 for a less restrictive version requiring more axioms. (Revised by GG, 20-Jan-2024.) Avoid ax-6 1967, ax-7 2007, ax-12 2177. (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 2715 . . 3 (𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑} ↔ [𝑧 / 𝑦]∀𝑥𝜑)
2 sbal 2169 . . . 4 ([𝑧 / 𝑦]∀𝑥𝜑 ↔ ∀𝑥[𝑧 / 𝑦]𝜑)
3 nfa1 2151 . . . 4 𝑥𝑥[𝑧 / 𝑦]𝜑
42, 3nfxfr 1853 . . 3 𝑥[𝑧 / 𝑦]∀𝑥𝜑
51, 4nfxfr 1853 . 2 𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑}
65nfci 2893 1 𝑥{𝑦 ∣ ∀𝑥𝜑}
Colors of variables: wff setvar class
Syntax hints:  wal 1538  [wsb 2064  wcel 2108  {cab 2714  wnfc 2890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-10 2141  ax-11 2157
This theorem depends on definitions:  df-bi 207  df-or 849  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-nfc 2892
This theorem is referenced by:  nfopd  4890  nfimad  6087  nfiota1  6516  nffvd  6918  nfunidALT2  38970  nfunidALT  38971  nfopdALT  38972  setrec1  49210
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