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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 1930 | . 2 ⊢ (𝑦 = 𝑥 → (𝑦 = 𝑧 → 𝑥 = 𝑧)) | |
2 | 1 | equcoms 1934 | 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 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 |
This theorem is referenced by: equtrr 1936 equequ1 1939 equviniva 1947 ax6e 2238 equvini 2334 sbequi 2363 axsep 4708 bj-axsep 31981 |
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