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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 3651 | . 2 ⊢ ∩ x ∈ A B = {z ∣ ∀x ∈ A z ∈ B} | |
2 | nfiunxy.1 | . . . 4 ⊢ ℲyA | |
3 | nfiunxy.2 | . . . . 5 ⊢ ℲyB | |
4 | 3 | nfcri 2169 | . . . 4 ⊢ Ⅎy z ∈ B |
5 | 2, 4 | nfralxy 2354 | . . 3 ⊢ Ⅎy∀x ∈ A z ∈ B |
6 | 5 | nfab 2179 | . 2 ⊢ Ⅎy{z ∣ ∀x ∈ A z ∈ B} |
7 | 1, 6 | nfcxfr 2172 | 1 ⊢ Ⅎy∩ x ∈ A B |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1390 {cab 2023 Ⅎwnfc 2162 ∀wral 2300 ∩ ciin 3649 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-iin 3651 |
This theorem is referenced by: iinab 3709 |
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