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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 2171. (Revised by SN, 20-Sep-2023.) |
Ref | Expression |
---|---|
nfsab1 | ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-clab 2710 | . 2 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} ↔ [𝑦 / 𝑥]𝜑) | |
2 | nfs1v 2153 | . 2 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑 | |
3 | 1, 2 | nfxfr 1855 | 1 ⊢ Ⅎ𝑥 𝑦 ∈ {𝑥 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: Ⅎwnf 1785 [wsb 2067 ∈ wcel 2106 {cab 2709 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-10 2137 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 |
This theorem is referenced by: hbab1 2718 abbib 2804 clelabOLD 2880 nfab1 2905 ralab2 3693 rexab2 3695 eluniab 4923 elintabOLD 4963 opabex3d 7951 opabex3rd 7952 opabex3 7953 setindtrs 41754 rababg 42315 scottabf 42989 |
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