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Mirrors > Home > ILE Home > Th. List > nfiinya | GIF version |
Description: Bound-variable hypothesis builder for indexed intersection. (Contributed by Mario Carneiro, 25-Jan-2014.) |
Ref | Expression |
---|---|
nfiunya.1 | ⊢ Ⅎ𝑦𝐴 |
nfiunya.2 | ⊢ Ⅎ𝑦𝐵 |
Ref | Expression |
---|---|
nfiinya | ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iin 3731 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} | |
2 | nfiunya.1 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
3 | nfiunya.2 | . . . . 5 ⊢ Ⅎ𝑦𝐵 | |
4 | 3 | nfcri 2222 | . . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐵 |
5 | 2, 4 | nfralya 2416 | . . 3 ⊢ Ⅎ𝑦∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 |
6 | 5 | nfab 2233 | . 2 ⊢ Ⅎ𝑦{𝑧 ∣ ∀𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} |
7 | 1, 6 | nfcxfr 2225 | 1 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 1438 {cab 2074 Ⅎwnfc 2215 ∀wral 2359 ∩ ciin 3729 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-iin 3731 |
This theorem is referenced by: (None) |
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