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Mirrors > Home > MPE Home > Th. List > nfin | Structured version Visualization version GIF version |
Description: Bound-variable hypothesis builder for the intersection of classes. (Contributed by NM, 15-Sep-2003.) (Revised by Mario Carneiro, 14-Oct-2016.) |
Ref | Expression |
---|---|
nfin.1 | ⊢ Ⅎ𝑥𝐴 |
nfin.2 | ⊢ Ⅎ𝑥𝐵 |
Ref | Expression |
---|---|
nfin | ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfin5 3941 | . 2 ⊢ (𝐴 ∩ 𝐵) = {𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵} | |
2 | nfin.2 | . . . 4 ⊢ Ⅎ𝑥𝐵 | |
3 | 2 | nfcri 2968 | . . 3 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 |
4 | nfin.1 | . . 3 ⊢ Ⅎ𝑥𝐴 | |
5 | 3, 4 | nfrabw 3383 | . 2 ⊢ Ⅎ𝑥{𝑦 ∈ 𝐴 ∣ 𝑦 ∈ 𝐵} |
6 | 1, 5 | nfcxfr 2972 | 1 ⊢ Ⅎ𝑥(𝐴 ∩ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 Ⅎwnfc 2958 {crab 3139 ∩ cin 3932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-in 3940 |
This theorem is referenced by: csbin 4388 iunxdif3 5008 disjxun 5055 nfres 5848 nfpred 6146 cp 9308 tskwe 9367 iunconn 21964 ptclsg 22151 restmetu 23107 limciun 24419 disjunsn 30272 ordtconnlem1 31066 esum2d 31251 finminlem 33563 bj-rcleqf 34234 mbfposadd 34820 iunconnlem2 41146 inn0f 41212 disjrnmpt2 41325 disjinfi 41330 fsumiunss 41732 stoweidlem57 42219 fourierdlem80 42348 sge0iunmptlemre 42574 iundjiun 42619 pimiooltgt 42866 smflim 42930 smfpimcclem 42958 smfpimcc 42959 |
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