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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 2127 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) |
4 | 3eqtrd.3 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | 3, 4 | eqtr2d 2128 | 1 ⊢ (𝜑 → 𝐷 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1388 ax-gen 1390 ax-4 1452 ax-17 1471 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-cleq 2088 |
This theorem is referenced by: nnanq0 7114 1idprl 7246 1idpru 7247 axcnre 7513 fseq1p1m1 9657 expmulzap 10116 expubnd 10127 subsq 10176 bcm1k 10283 bcpasc 10289 crim 10407 rereb 10412 fsumparts 11013 isumshft 11033 geosergap 11049 efsub 11120 sincossq 11188 efieq1re 11210 bezoutlema 11415 bezoutlemb 11416 eucalg 11468 phiprmpw 11625 strsetsid 11676 setsslid 11693 |
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