Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > s3eq2 | Structured version Visualization version GIF version |
Description: Equality theorem for a length 3 word for the second symbol. (Contributed by AV, 4-Jan-2022.) |
Ref | Expression |
---|---|
s3eq2 | ⊢ (𝐵 = 𝐷 → 〈“𝐴𝐵𝐶”〉 = 〈“𝐴𝐷𝐶”〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqidd 2737 | . 2 ⊢ (𝐵 = 𝐷 → 𝐴 = 𝐴) | |
2 | id 22 | . 2 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷) | |
3 | eqidd 2737 | . 2 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐶) | |
4 | 1, 2, 3 | s3eqd 14394 | 1 ⊢ (𝐵 = 𝐷 → 〈“𝐴𝐵𝐶”〉 = 〈“𝐴𝐷𝐶”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 〈“cs3 14372 |
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 2018 ax-8 2114 ax-9 2122 ax-ext 2708 |
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 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4224 df-if 4426 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-iota 6316 df-fv 6366 df-ov 7194 df-s1 14118 df-s2 14378 df-s3 14379 |
This theorem is referenced by: tgcgrxfr 26563 isperp2 26760 elwwlks2ons3 27993 frgr2wwlk1 28366 frgr2wwlkeqm 28368 fusgr2wsp2nb 28371 |
Copyright terms: Public domain | W3C validator |