Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 4747 | . 2 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐶)) |
3 | coeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | coeq2d 4748 | . 2 ⊢ (𝜑 → (𝐵 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
5 | 2, 4 | eqtrd 2190 | 1 ⊢ (𝜑 → (𝐴 ∘ 𝐶) = (𝐵 ∘ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = 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) |
Copyright terms: Public domain | W3C validator |