Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > difeq12d | GIF version |
Description: Equality deduction for class difference. (Contributed by FL, 29-May-2014.) |
Ref | Expression |
---|---|
difeq12d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
difeq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
difeq12d | ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difeq12d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | difeq1d 3163 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) |
3 | difeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | difeq2d 3164 | . 2 ⊢ (𝜑 → (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)) |
5 | 2, 4 | eqtrd 2150 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1316 ∖ cdif 3038 |
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-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-tru 1319 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rab 2402 df-dif 3043 |
This theorem is referenced by: undifexmid 4087 exmidundif 4099 exmidundifim 4100 |
Copyright terms: Public domain | W3C validator |