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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 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 14750 | 1 ⊢ (𝐵 = 𝐷 → 〈“𝐴𝐵𝐶”〉 = 〈“𝐴𝐷𝐶”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1541 〈“cs3 14728 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2714 df-cleq 2728 df-clel 2814 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-nul 4282 df-if 4486 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-iota 6446 df-fv 6502 df-ov 7357 df-s1 14481 df-s2 14734 df-s3 14735 |
This theorem is referenced by: tgcgrxfr 27358 isperp2 27555 elwwlks2ons3 28798 frgr2wwlk1 29171 frgr2wwlkeqm 29173 fusgr2wsp2nb 29176 |
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