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Mirrors > Home > ILE Home > Th. List > coeq2d | GIF version |
Description: Equality deduction for composition of two classes. (Contributed by NM, 16-Nov-2000.) |
Ref | Expression |
---|---|
coeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
coeq2d | ⊢ (𝜑 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | coeq2 4692 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ∘ 𝐴) = (𝐶 ∘ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 ∘ ccom 4538 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-in 3072 df-ss 3079 df-br 3925 df-opab 3985 df-co 4543 |
This theorem is referenced by: coeq12d 4698 relcoi1 5065 f1ococnv1 5389 funcoeqres 5391 fcof1o 5683 foeqcnvco 5684 mapen 6733 hashfacen 10572 |
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