![]() |
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 3132 | . 2 ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐶)) |
3 | difeq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | difeq2d 3133 | . 2 ⊢ (𝜑 → (𝐵 ∖ 𝐶) = (𝐵 ∖ 𝐷)) |
5 | 2, 4 | eqtrd 2127 | 1 ⊢ (𝜑 → (𝐴 ∖ 𝐶) = (𝐵 ∖ 𝐷)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1296 ∖ cdif 3010 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 |
This theorem depends on definitions: df-bi 116 df-tru 1299 df-nf 1402 df-sb 1700 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ral 2375 df-rab 2379 df-dif 3015 |
This theorem is referenced by: undifexmid 4049 exmidundif 4058 exmidundifim 4059 |
Copyright terms: Public domain | W3C validator |