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Mirrors > Home > ILE Home > Th. List > 3eqtrrd | GIF version |
Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) (Proof shortened by Andrew Salmon, 25-May-2011.) |
Ref | Expression |
---|---|
3eqtrd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
3eqtrd.2 | ⊢ (𝜑 → 𝐵 = 𝐶) |
3eqtrd.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
3eqtrrd | ⊢ (𝜑 → 𝐷 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtrd.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 3eqtrd.2 | . . 3 ⊢ (𝜑 → 𝐵 = 𝐶) | |
3 | 1, 2 | eqtrd 2072 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) |
4 | 3eqtrd.3 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | 3, 4 | eqtr2d 2073 | 1 ⊢ (𝜑 → 𝐷 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1336 ax-gen 1338 ax-4 1400 ax-17 1419 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-cleq 2033 |
This theorem is referenced by: nnanq0 6556 1idprl 6688 1idpru 6689 axcnre 6955 fseq1p1m1 8956 expmulzap 9301 expubnd 9311 subsq 9358 crim 9458 rereb 9463 |
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