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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 4810 | . 2 ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐶, 𝐸〉) |
3 | oteq123d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | oteq2d 4811 | . 2 ⊢ (𝜑 → 〈𝐵, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐸〉) |
5 | oteq123d.3 | . . 3 ⊢ (𝜑 → 𝐸 = 𝐹) | |
6 | 5 | oteq3d 4812 | . 2 ⊢ (𝜑 → 〈𝐵, 𝐷, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
7 | 2, 4, 6 | 3eqtrd 2782 | 1 ⊢ (𝜑 → 〈𝐴, 𝐶, 𝐸〉 = 〈𝐵, 𝐷, 𝐹〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 〈cotp 4563 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-ext 2709 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2072 df-clab 2716 df-cleq 2730 df-clel 2817 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-nul 4252 df-if 4454 df-sn 4556 df-pr 4558 df-op 4562 df-ot 4564 |
This theorem is referenced by: idaval 17588 coaval 17598 matval 21332 msrval 33236 mclsax 33267 elmpps 33271 mthmpps 33280 ackval0012 45736 ackval1012 45737 ackval2012 45738 ackval3012 45739 mndtcval 46065 |
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