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Mirrors > Home > ILE Home > Th. List > 3eqtr2rd | GIF version |
Description: A deduction from three chained equalities. (Contributed by NM, 4-Aug-2006.) |
Ref | Expression |
---|---|
3eqtr2d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
3eqtr2d.2 | ⊢ (𝜑 → 𝐶 = 𝐵) |
3eqtr2d.3 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
3eqtr2rd | ⊢ (𝜑 → 𝐷 = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtr2d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 3eqtr2d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐵) | |
3 | 1, 2 | eqtr4d 2116 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) |
4 | 3eqtr2d.3 | . 2 ⊢ (𝜑 → 𝐶 = 𝐷) | |
5 | 3, 4 | eqtr2d 2114 | 1 ⊢ (𝜑 → 𝐷 = 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1376 ax-gen 1378 ax-4 1440 ax-17 1459 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-cleq 2074 |
This theorem is referenced by: prarloclemlo 6684 recexgt0sr 6950 xp1d2m1eqxm1d2 8283 qnegmod 9371 modqeqmodmin 9396 faclbnd2 9669 cjmulval 9775 qredeu 10479 |
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