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Mirrors > Home > ILE Home > Th. List > equtr2 | GIF version |
Description: A transitive law for equality. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
equtr2 | ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equtrr 1596 | . . 3 ⊢ (𝑧 = 𝑦 → (𝑥 = 𝑧 → 𝑥 = 𝑦)) | |
2 | 1 | equcoms 1594 | . 2 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑧 → 𝑥 = 𝑦)) |
3 | 2 | impcom 116 | 1 ⊢ ((𝑥 = 𝑧 ∧ 𝑦 = 𝑧) → 𝑥 = 𝑦) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-gen 1338 ax-ie2 1383 ax-8 1395 ax-17 1419 ax-i9 1423 |
This theorem depends on definitions: df-bi 110 |
This theorem is referenced by: mo23 1941 euequ1 1995 |
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