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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 4807 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐶, 𝐸〉) |
3 | oteq123d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | oteq2d 4808 | . 2 ⊢ (𝜑 → 〈𝐵, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐸〉) |
5 | oteq123d.3 | . . 3 ⊢ (𝜑 → 𝐸 = 𝐹) | |
6 | 5 | oteq3d 4809 | . 2 ⊢ (𝜑 → 〈𝐵, 𝐷, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
7 | 2, 4, 6 | 3eqtrd 2857 | 1 ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 〈cotp 4565 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-ot 4566 |
This theorem is referenced by: idaval 17306 coaval 17316 matval 20948 msrval 32682 mclsax 32713 elmpps 32717 mthmpps 32726 |
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