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Mirrors > Home > ILE Home > Th. List > 3eqtr3rd | GIF version |
Description: A deduction from three chained equalities. (Contributed by NM, 14-Jan-2006.) |
Ref | Expression |
---|---|
3eqtr3d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
3eqtr3d.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
3eqtr3d.3 | ⊢ (𝜑 → 𝐵 = 𝐷) |
Ref | Expression |
---|---|
3eqtr3rd | ⊢ (𝜑 → 𝐷 = 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 3eqtr3d.3 | . 2 ⊢ (𝜑 → 𝐵 = 𝐷) | |
2 | 3eqtr3d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
3 | 3eqtr3d.2 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐶) | |
4 | 2, 3 | eqtr3d 2115 | . 2 ⊢ (𝜑 → 𝐵 = 𝐶) |
5 | 1, 4 | eqtr3d 2115 | 1 ⊢ (𝜑 → 𝐷 = 𝐶) |
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: fcofo 5444 fcof1o 5449 nnaword 6107 pn0sr 6948 negeu 7299 add20 7578 2halves 8260 bcnn 9684 bcpasc 9693 resqrexlemover 9896 gcdid 10377 |
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