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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 2796 | . 2 ⊢ (𝐵 = 𝐷 → 𝐴 = 𝐴) | |
2 | id 22 | . 2 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷) | |
3 | eqidd 2796 | . 2 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐶) | |
4 | 1, 2, 3 | s3eqd 14067 | 1 ⊢ (𝐵 = 𝐷 → 〈“𝐴𝐵𝐶”〉 = 〈“𝐴𝐷𝐶”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1522 〈“cs3 14045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-rex 3111 df-rab 3114 df-v 3439 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-nul 4216 df-if 4386 df-sn 4477 df-pr 4479 df-op 4483 df-uni 4750 df-br 4967 df-iota 6194 df-fv 6238 df-ov 7024 df-s1 13799 df-s2 14051 df-s3 14052 |
This theorem is referenced by: tgcgrxfr 25991 isperp2 26188 elwwlks2ons3 27426 frgr2wwlk1 27805 frgr2wwlkeqm 27807 fusgr2wsp2nb 27810 |
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