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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 2822 | . 2 ⊢ (𝐵 = 𝐷 → 𝐴 = 𝐴) | |
2 | id 22 | . 2 ⊢ (𝐵 = 𝐷 → 𝐵 = 𝐷) | |
3 | eqidd 2822 | . 2 ⊢ (𝐵 = 𝐷 → 𝐶 = 𝐶) | |
4 | 1, 2, 3 | s3eqd 14220 | 1 ⊢ (𝐵 = 𝐷 → 〈“𝐴𝐵𝐶”〉 = 〈“𝐴𝐷𝐶”〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 〈“cs3 14198 |
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 2156 ax-12 2172 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-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-iota 6309 df-fv 6358 df-ov 7153 df-s1 13944 df-s2 14204 df-s3 14205 |
This theorem is referenced by: tgcgrxfr 26298 isperp2 26495 elwwlks2ons3 27728 frgr2wwlk1 28102 frgr2wwlkeqm 28104 fusgr2wsp2nb 28107 |
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