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| Mirrors > Home > MPE Home > Th. List > nfii1 | Structured version Visualization version GIF version | ||
| Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by NM, 15-Oct-2003.) |
| Ref | Expression |
|---|---|
| nfii1 | ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-iin 4994 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
| 2 | nfra1 3284 | . . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
| 3 | 2 | nfab 2911 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
| 4 | 1, 3 | nfcxfr 2903 | 1 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 {cab 2714 Ⅎwnfc 2890 ∀wral 3061 ∩ ciin 4992 |
| 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-iin 4994 |
| This theorem is referenced by: dmiin 5964 scott0 9926 gruiin 10850 zarclsiin 33870 iinssiin 45134 iooiinicc 45555 iooiinioc 45569 fnlimfvre 45689 fnlimabslt 45694 meaiininclem 46501 hspdifhsp 46631 smflimlem2 46787 smflim 46792 smflimmpt 46825 smfsuplem1 46826 smfsupmpt 46830 smfsupxr 46831 smfinflem 46832 smfinfmpt 46834 smflimsuplem7 46841 smflimsuplem8 46842 smflimsupmpt 46844 smfliminfmpt 46847 fsupdm 46857 finfdm 46861 |
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