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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 35704 | . 2 ⊢ (𝐴 = 𝐵 → ≀ 𝐴 = ≀ 𝐵) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ ≀ 𝐴 = ≀ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ≀ ccoss 35486 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-br 5060 df-opab 5122 df-coss 35692 |
This theorem is referenced by: br1cossinres 35720 br1cossxrnres 35721 cosscnvid 35754 |
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