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Mirrors > Home > ILE Home > Th. List > axext3 | GIF version |
Description: A generalization of the Axiom of Extensionality in which 𝑥 and 𝑦 need not be distinct. (Contributed by NM, 15-Sep-1993.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) |
Ref | Expression |
---|---|
axext3 | ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elequ2 1691 | . . . . 5 ⊢ (𝑤 = 𝑥 → (𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑥)) | |
2 | 1 | bibi1d 232 | . . . 4 ⊢ (𝑤 = 𝑥 → ((𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ (𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) |
3 | 2 | albidv 1796 | . . 3 ⊢ (𝑤 = 𝑥 → (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) ↔ ∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦))) |
4 | equequ1 1688 | . . 3 ⊢ (𝑤 = 𝑥 → (𝑤 = 𝑦 ↔ 𝑥 = 𝑦)) | |
5 | 3, 4 | imbi12d 233 | . 2 ⊢ (𝑤 = 𝑥 → ((∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦) ↔ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦))) |
6 | ax-ext 2121 | . 2 ⊢ (∀𝑧(𝑧 ∈ 𝑤 ↔ 𝑧 ∈ 𝑦) → 𝑤 = 𝑦) | |
7 | 5, 6 | chvarv 1909 | 1 ⊢ (∀𝑧(𝑧 ∈ 𝑥 ↔ 𝑧 ∈ 𝑦) → 𝑥 = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1329 |
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-5 1423 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 |
This theorem is referenced by: axext4 2123 |
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