|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > equtr | Structured version Visualization version GIF version | ||
| Description: A transitive law for equality. (Contributed by NM, 23-Aug-1993.) | 
| Ref | Expression | 
|---|---|
| equtr | ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ax7 2014 | . 2 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
| 2 | 1 | equcoms 2018 | 1 ⊢ (𝑥 = 𝑦 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1779 | 
| This theorem is referenced by: equtrr 2020 equequ1 2023 equvinva 2028 ax6e 2387 equvini 2459 axprlem3OLD 5427 | 
| Copyright terms: Public domain | W3C validator |