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| Mirrors > Home > MPE Home > Th. List > nfpw | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for power class. (Contributed by NM, 28-Oct-2003.) (Revised by Mario Carneiro, 13-Oct-2016.) |
| Ref | Expression |
|---|---|
| nfpw.1 | ⊢ Ⅎ𝑥𝐴 |
| Ref | Expression |
|---|---|
| nfpw | ⊢ Ⅎ𝑥𝒫 𝐴 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-pw 4602 | . 2 ⊢ 𝒫 𝐴 = {𝑦 ∣ 𝑦 ⊆ 𝐴} | |
| 2 | nfcv 2905 | . . . 4 ⊢ Ⅎ𝑥𝑦 | |
| 3 | nfpw.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
| 4 | 2, 3 | nfss 3976 | . . 3 ⊢ Ⅎ𝑥 𝑦 ⊆ 𝐴 |
| 5 | 4 | nfab 2911 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ 𝑦 ⊆ 𝐴} |
| 6 | 1, 5 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥𝒫 𝐴 |
| Colors of variables: wff setvar class |
| Syntax hints: {cab 2714 Ⅎwnfc 2890 ⊆ wss 3951 𝒫 cpw 4600 |
| 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-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 |
| 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 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-ss 3968 df-pw 4602 |
| This theorem is referenced by: esum2d 34094 ldsysgenld 34161 stoweidlem57 46072 sge0iunmptlemre 46430 nfafv2 47230 |
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