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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 2023 | . 2 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
2 | 1 | equcoms 2027 | 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 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 |
This theorem is referenced by: equtrr 2029 equequ1 2032 equvinva 2037 ax6e 2390 equvini 2466 equviniOLD 2467 sbequiALT 2572 axprlem3 5291 |
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