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Mirrors > Home > ILE Home > Th. List > nfint | GIF version |
Description: Bound-variable hypothesis builder for intersection. (Contributed by NM, 2-Feb-1997.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
nfint.1 | ⊢ Ⅎ𝑥𝐴 |
Ref | Expression |
---|---|
nfint | ⊢ Ⅎ𝑥∩ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfint2 3848 | . 2 ⊢ ∩ 𝐴 = {𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} | |
2 | nfint.1 | . . . 4 ⊢ Ⅎ𝑥𝐴 | |
3 | nfv 1528 | . . . 4 ⊢ Ⅎ𝑥 𝑦 ∈ 𝑧 | |
4 | 2, 3 | nfralxy 2515 | . . 3 ⊢ Ⅎ𝑥∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧 |
5 | 4 | nfab 2324 | . 2 ⊢ Ⅎ𝑥{𝑦 ∣ ∀𝑧 ∈ 𝐴 𝑦 ∈ 𝑧} |
6 | 1, 5 | nfcxfr 2316 | 1 ⊢ Ⅎ𝑥∩ 𝐴 |
Colors of variables: wff set class |
Syntax hints: {cab 2163 Ⅎwnfc 2306 ∀wral 2455 ∩ cint 3846 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-tru 1356 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-int 3847 |
This theorem is referenced by: (None) |
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