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Mirrors > Home > ILE Home > Th. List > nfii1 | 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 3848 | . 2 ⊢ ∩ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} | |
2 | nfra1 2485 | . . 3 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵 | |
3 | 2 | nfab 2301 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} |
4 | 1, 3 | nfcxfr 2293 | 1 ⊢ Ⅎ𝑥∩ 𝑥 ∈ 𝐴 𝐵 |
Colors of variables: wff set class |
Syntax hints: ∈ wcel 2125 {cab 2140 Ⅎwnfc 2283 ∀wral 2432 ∩ ciin 3846 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1481 ax-10 1482 ax-11 1483 ax-i12 1484 ax-bndl 1486 ax-4 1487 ax-17 1503 ax-i9 1507 ax-ial 1511 ax-i5r 1512 ax-ext 2136 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1740 df-clab 2141 df-cleq 2147 df-clel 2150 df-nfc 2285 df-ral 2437 df-iin 3848 |
This theorem is referenced by: dmiin 4825 |
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