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Mirrors > Home > ILE Home > Th. List > coeq12i | GIF version |
Description: Equality inference for composition of two classes. (Contributed by FL, 7-Jun-2012.) |
Ref | Expression |
---|---|
coeq12i.1 | ⊢ 𝐴 = 𝐵 |
coeq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
coeq12i | ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq12i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | 1 | coeq1i 4745 | . 2 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶) |
3 | coeq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | coeq2i 4746 | . 2 ⊢ (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
5 | 2, 4 | eqtri 2178 | 1 ⊢ (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷) |
Colors of variables: wff set class |
Syntax hints: = wceq 1335 ∘ ccom 4590 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2139 |
This theorem depends on definitions: df-bi 116 df-nf 1441 df-sb 1743 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-in 3108 df-ss 3115 df-br 3966 df-opab 4026 df-co 4595 |
This theorem is referenced by: (None) |
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