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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 4999 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
2 | nfra1 3282 | . . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfab 2909 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
4 | 1, 3 | nfcxfr 2901 | 1 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2106 {cab 2712 Ⅎwnfc 2888 ∀wral 3059 ∩ ciin 4997 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-iin 4999 |
This theorem is referenced by: dmiin 5967 scott0 9924 gruiin 10848 zarclsiin 33832 iinssiin 45069 iooiinicc 45495 iooiinioc 45509 fnlimfvre 45630 fnlimabslt 45635 meaiininclem 46442 hspdifhsp 46572 smflimlem2 46728 smflim 46733 smflimmpt 46766 smfsuplem1 46767 smfsupmpt 46771 smfsupxr 46772 smfinflem 46773 smfinfmpt 46775 smflimsuplem7 46782 smflimsuplem8 46783 smflimsupmpt 46785 smfliminfmpt 46788 fsupdm 46798 finfdm 46802 |
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