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Mirrors > Home > MPE Home > Th. List > nfaba1 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
nfaba1 | ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} |
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} |
Colors of variables: wff setvar class |
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 |
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