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Mirrors > Home > MPE Home > Th. List > nfsab1 | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 11-Aug-2016.) Remove use of ax-12 2167. (Revised by SN, 20-Sep-2023.) |
Ref | Expression |
---|---|
nfsab1 | ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2797 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
2 | nfs1v 2264 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
3 | 1, 2 | nfxfr 1844 | 1 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnf 1775 [wsb 2060 ∈ wcel 2105 {cab 2796 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-10 2136 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 |
This theorem is referenced by: abbi 2885 clelab 2955 nfab1 2976 ralab2 3685 ralab2OLD 3686 rexab2 3688 rexab2OLD 3689 eluniab 4841 elintab 4878 opabex3d 7655 opabex3rd 7656 opabex3 7657 setindtrs 39500 rababg 39811 scottabf 40453 |
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