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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 5201 | . 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 1475 ∘ ccom 5042 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-in 3547 df-ss 3554 df-br 4584 df-opab 4644 df-co 5047 |
This theorem is referenced by: coeq12i 5207 cocnvcnv1 5563 hashgval 12982 imasdsval2 15999 prds1 18437 pf1mpf 19537 upxp 21236 uptx 21238 hoico2 28000 hoid1ri 28033 nmopcoadj2i 28345 pjclem3 28440 erdsze2lem2 30440 pprodcnveq 31160 diblss 35477 cononrel2 36920 trclubgNEW 36944 cortrcltrcl 37051 corclrtrcl 37052 cortrclrcl 37054 cotrclrtrcl 37055 cortrclrtrcl 37056 neicvgbex 37430 neicvgnvo 37433 dvsinax 38801 |
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