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Mirrors > Home > MPE Home > Th. List > oteq123d | Structured version Visualization version GIF version |
Description: Equality deduction for ordered triples. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
oteq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
oteq123d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
oteq123d.3 | ⊢ (𝜑 → 𝐸 = 𝐹) |
Ref | Expression |
---|---|
oteq123d | ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oteq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | oteq1d 4648 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐶, 𝐸〉) |
3 | oteq123d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | oteq2d 4649 | . 2 ⊢ (𝜑 → 〈𝐵, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐸〉) |
5 | oteq123d.3 | . . 3 ⊢ (𝜑 → 𝐸 = 𝐹) | |
6 | 5 | oteq3d 4650 | . 2 ⊢ (𝜑 → 〈𝐵, 𝐷, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
7 | 2, 4, 6 | 3eqtrd 2817 | 1 ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 〈cotp 4405 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-rab 3098 df-v 3399 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-nul 4141 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-ot 4406 |
This theorem is referenced by: idaval 17093 coaval 17103 matval 20621 msrval 32048 mclsax 32079 elmpps 32083 mthmpps 32092 |
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