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| 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 2738 | . 2 ⊢ (𝐵 = 𝐷 → 𝐴 = 𝐴) | |
| 2 | id 22 | . 2 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷) | |
| 3 | eqidd 2738 | . 2 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐶) | |
| 4 | 1, 2, 3 | s3eqd 14903 | 1 ⊢ (𝐵 = 𝐷 → 〈“𝐴𝐵𝐶”〉 = 〈“𝐴𝐷𝐶”〉) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1540 〈“cs3 14881 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-s1 14634 df-s2 14887 df-s3 14888 | 
| This theorem is referenced by: tgcgrxfr 28526 isperp2 28723 elwwlks2ons3 29975 frgr2wwlk1 30348 frgr2wwlkeqm 30350 fusgr2wsp2nb 30353 | 
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