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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 4772 | . 2 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)) |
3 | coeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | coeq2d 4773 | . 2 ⊢ (𝜑 → (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
5 | 2, 4 | eqtrd 2203 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1348 ∘ ccom 4615 |
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 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-in 3127 df-ss 3134 df-br 3990 df-opab 4051 df-co 4620 |
This theorem is referenced by: (None) |
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