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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 2177. (Revised by SN, 20-Sep-2023.) | 
| Ref | Expression | 
|---|---|
| nfsab1 | ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | df-clab 2715 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
| 2 | nfs1v 2156 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
| 3 | 1, 2 | nfxfr 1853 | 1 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} | 
| Colors of variables: wff setvar class | 
| Syntax hints: Ⅎwnf 1783 [wsb 2064 ∈ wcel 2108 {cab 2714 | 
| 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-6 1967 ax-7 2007 ax-10 2141 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 | 
| This theorem is referenced by: hbab1 2723 abbib 2811 nfab1 2907 ralab2 3703 rexab2 3705 eluniab 4921 elintabOLD 4959 opabex3d 7990 opabex3rd 7991 opabex3 7992 setindtrs 43037 rababg 43587 scottabf 44259 | 
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