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Mirrors > Home > ILE Home > Th. List > coeq12d | GIF version |
Description: Equality deduction for composition of two classes. (Contributed by FL, 7-Jun-2012.) |
Ref | Expression |
---|---|
coeq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
coeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
coeq12d | ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | coeq12d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | coeq1d 4515 | . 2 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)) |
3 | coeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | coeq2d 4516 | . 2 ⊢ (𝜑 → (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
5 | 2, 4 | eqtrd 2113 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1284 ∘ ccom 4367 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 |
This theorem depends on definitions: df-bi 115 df-nf 1390 df-sb 1686 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-in 2979 df-ss 2986 df-br 3786 df-opab 3840 df-co 4372 |
This theorem is referenced by: (None) |
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