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Mirrors > Home > ILE Home > Th. List > dveel1 | GIF version |
Description: Quantifier introduction when one pair of variables is disjoint. (Contributed by NM, 2-Jan-2002.) |
Ref | Expression |
---|---|
dveel1 | ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-17 1513 | . 2 ⊢ (𝑤 ∈ 𝑧 → ∀𝑥 𝑤 ∈ 𝑧) | |
2 | ax-17 1513 | . 2 ⊢ (𝑦 ∈ 𝑧 → ∀𝑤 𝑦 ∈ 𝑧) | |
3 | elequ1 2139 | . 2 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) | |
4 | 1, 2, 3 | dvelimf 2002 | 1 ⊢ (¬ ∀𝑥 𝑥 = 𝑦 → (𝑦 ∈ 𝑧 → ∀𝑥 𝑦 ∈ 𝑧)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∀wal 1340 |
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-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 |
This theorem depends on definitions: df-bi 116 df-nf 1448 df-sb 1750 |
This theorem is referenced by: (None) |
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