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Mathbox for Peter Mazsa |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > cosseqi | Structured version Visualization version GIF version |
Description: Equality theorem for the classes of cosets by 𝐴 and 𝐵, inference form. (Contributed by Peter Mazsa, 9-Jan-2018.) |
Ref | Expression |
---|---|
cosseqi.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
cosseqi | ⊢ ≀ 𝐴 = ≀ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cosseqi.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | cosseq 37808 | . 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ≀ 𝐴 = ≀ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ≀ ccoss 37555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-br 5142 df-opab 5204 df-coss 37793 |
This theorem is referenced by: br1cossinres 37829 br1cossxrnres 37830 cosscnvid 37863 |
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