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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 4626 | . . 3 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴〉 = 〈𝐶, 𝐵〉) | |
2 | 1 | opeq1d 4631 | . 2 ⊢ (𝐴 = 𝐵 → 〈〈𝐶, 𝐴〉, 𝐷〉 = 〈〈𝐶, 𝐵〉, 𝐷〉) |
3 | df-ot 4408 | . 2 ⊢ 〈𝐶, 𝐴, 𝐷〉 = 〈〈𝐶, 𝐴〉, 𝐷〉 | |
4 | df-ot 4408 | . 2 ⊢ 〈𝐶, 𝐵, 𝐷〉 = 〈〈𝐶, 𝐵〉, 𝐷〉 | |
5 | 2, 3, 4 | 3eqtr4g 2886 | 1 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 〈cop 4405 〈cotp 4407 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-ot 4408 |
This theorem is referenced by: oteq2d 4638 efgi 18490 efgtf 18493 efgtval 18494 |
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