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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 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.) |
Ref | Expression |
---|---|
nfaba1 | ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} |
Step | Hyp | Ref | 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 ⊢ Ⅎ𝑥∀𝑥[𝑧 / 𝑦]𝜑 | |
4 | 2, 3 | nfxfr 1850 | . . 3 ⊢ Ⅎ𝑥[𝑧 / 𝑦]∀𝑥𝜑 |
5 | 1, 4 | nfxfr 1850 | . 2 ⊢ Ⅎ𝑥 𝑧 ∈ {𝑦 ∣ ∀𝑥𝜑} |
6 | 5 | nfci 2891 | 1 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∀wal 1535 [wsb 2062 ∈ wcel 2106 {cab 2712 Ⅎwnfc 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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