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Mirrors > Home > ILE Home > Th. List > coeq2i | GIF version |
Description: Equality inference for composition of two classes. (Contributed by NM, 16-Nov-2000.) |
Ref | Expression |
---|---|
coeq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
coeq2i | ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | coeq2 4635 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1299 ∘ ccom 4481 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 |
This theorem depends on definitions: df-bi 116 df-nf 1405 df-sb 1704 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-in 3027 df-ss 3034 df-br 3876 df-opab 3930 df-co 4486 |
This theorem is referenced by: coeq12i 4640 cocnvcnv2 4986 co01 4989 cocnvres 4999 fcoi1 5239 dftpos2 6088 tposco 6102 |
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