![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > oteq2 | Structured version Visualization version GIF version |
Description: Equality theorem for ordered triples. (Contributed by NM, 3-Apr-2015.) |
Ref | Expression |
---|---|
oteq2 | ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq2 4870 | . . 3 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴⟩ = ⟨𝐶, 𝐵⟩) | |
2 | 1 | opeq1d 4875 | . 2 ⊢ (𝐴 = 𝐵 → ⟨⟨𝐶, 𝐴⟩, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷⟩) |
3 | df-ot 4633 | . 2 ⊢ ⟨𝐶, 𝐴, 𝐷⟩ = ⟨⟨𝐶, 𝐴⟩, 𝐷⟩ | |
4 | df-ot 4633 | . 2 ⊢ ⟨𝐶, 𝐵, 𝐷⟩ = ⟨⟨𝐶, 𝐵⟩, 𝐷⟩ | |
5 | 2, 3, 4 | 3eqtr4g 2793 | 1 ⊢ (𝐴 = 𝐵 → ⟨𝐶, 𝐴, 𝐷⟩ = ⟨𝐶, 𝐵, 𝐷⟩) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ⟨cop 4630 ⟨cotp 4632 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2699 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-ot 4633 |
This theorem is referenced by: oteq2d 4882 frxp3 8150 xpord3pred 8151 efgi 19667 efgtf 19670 efgtval 19671 |
Copyright terms: Public domain | W3C validator |