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Mirrors > Home > ILE Home > Th. List > nfiinxy | GIF version |
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) |
Ref | Expression |
---|---|
nfiunxy.1 | ⊢ Ⅎ𝑦𝐴 |
nfiunxy.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfiinxy | ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 3786 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfiunxy.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfiunxy.2 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | 3 | nfcri 2252 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | 2, 4 | nfralxy 2448 | . . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 |
6 | 5 | nfab 2263 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2255 | 1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1465 {cab 2103 Ⅎwnfc 2245 ∀wral 2393 ∩ ciin 3784 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-iin 3786 |
This theorem is referenced by: iinab 3844 |
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