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Mirrors > Home > MPE Home > Th. List > coeq1i | Structured version Visualization version GIF version |
Description: Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.) |
Ref | Expression |
---|---|
coeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
coeq1i | ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | coeq1 5854 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∘ ccom 5678 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-in 3953 df-ss 3963 df-br 5147 df-opab 5209 df-co 5683 |
This theorem is referenced by: coeq12i 5860 cocnvcnv1 6252 ttrclco 9708 hashgval 14288 imasdsval2 17457 prds1 20125 pf1mpf 21852 upxp 23108 uptx 23110 hoico2 30987 hoid1ri 31020 nmopcoadj2i 31332 pjclem3 31427 cycpmconjslem1 32290 cycpmconjs 32292 cyc3conja 32293 erdsze2lem2 34132 pprodcnveq 34792 diblss 39978 cononrel2 42278 trclubgNEW 42301 cortrcltrcl 42423 corclrtrcl 42424 cortrclrcl 42426 cotrclrtrcl 42427 cortrclrtrcl 42428 neicvgbex 42795 neicvgnvo 42798 dvsinax 44563 |
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