Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > oteq2d | 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) |
Ref | Expression |
---|---|
oteq2d | ⊢ (𝜑 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | oteq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | oteq2 4806 | . 2 ⊢ (𝐴 = 𝐵 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → 〈𝐶, 𝐴, 𝐷〉 = 〈𝐶, 𝐵, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 〈cotp 4568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4561 df-pr 4563 df-op 4567 df-ot 4569 |
This theorem is referenced by: oteq123d 4811 mapdh9a 38919 mapdh9aOLDN 38920 hdmap1eulem 38952 hdmap1eulemOLDN 38953 hdmapffval 38956 hdmapfval 38957 hdmapval2 38962 |
Copyright terms: Public domain | W3C validator |