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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 2169. (Revised by SN, 20-Sep-2023.) |
Ref | Expression |
---|---|
nfsab1 | ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2708 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
2 | nfs1v 2151 | . 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 2065 ∈ wcel 2104 {cab 2707 |
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 1911 ax-6 1969 ax-7 2009 ax-10 2135 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-ex 1780 df-nf 1784 df-sb 2066 df-clab 2708 |
This theorem is referenced by: hbab1 2716 abbib 2802 clelabOLD 2878 nfab1 2903 ralab2 3692 rexab2 3694 eluniab 4922 elintabOLD 4962 opabex3d 7954 opabex3rd 7955 opabex3 7956 setindtrs 42066 rababg 42627 scottabf 43301 |
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