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Mirrors > Home > ILE Home > Th. List > equvin | GIF version |
Description: A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
equvin | ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equvini 1738 | . 2 ⊢ (𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) | |
2 | ax-17 1506 | . . 3 ⊢ (𝑥 = 𝑦 → ∀𝑧 𝑥 = 𝑦) | |
3 | equtr 1689 | . . . 4 ⊢ (𝑥 = 𝑧 → (𝑧 = 𝑦 → 𝑥 = 𝑦)) | |
4 | 3 | imp 123 | . . 3 ⊢ ((𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦) |
5 | 2, 4 | exlimih 1573 | . 2 ⊢ (∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦) → 𝑥 = 𝑦) |
6 | 1, 5 | impbii 125 | 1 ⊢ (𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧 ∧ 𝑧 = 𝑦)) |
Colors of variables: wff set class |
Syntax hints: ∧ wa 103 ↔ wb 104 ∃wex 1472 |
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 1427 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-i12 1487 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 |
This theorem depends on definitions: df-bi 116 |
This theorem is referenced by: (None) |
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