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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 4922 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
2 | nfra1 3219 | . . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfab 2984 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
4 | 1, 3 | nfcxfr 2975 | 1 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 {cab 2799 Ⅎwnfc 2961 ∀wral 3138 ∩ ciin 4920 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-iin 4922 |
This theorem is referenced by: dmiin 5825 scott0 9315 gruiin 10232 iinssiin 41415 iooiinicc 41838 iooiinioc 41852 fnlimfvre 41975 fnlimabslt 41980 meaiininclem 42788 hspdifhsp 42918 smflimlem2 43068 smflim 43073 smflimmpt 43104 smfsuplem1 43105 smfsupmpt 43109 smfsupxr 43110 smfinflem 43111 smfinfmpt 43113 smflimsuplem7 43120 smflimsuplem8 43121 smflimsupmpt 43123 smfliminfmpt 43126 |
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