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Mirrors > Home > ILE Home > Th. List > ineq2d | GIF version |
Description: Equality deduction for intersection of two classes. (Contributed by NM, 10-Apr-1994.) |
Ref | Expression |
---|---|
ineq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
ineq2d | ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | ineq2 3181 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1287 ∩ cin 2985 |
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 663 ax-5 1379 ax-7 1380 ax-gen 1381 ax-ie1 1425 ax-ie2 1426 ax-8 1438 ax-10 1439 ax-11 1440 ax-i12 1441 ax-bndl 1442 ax-4 1443 ax-17 1462 ax-i9 1466 ax-ial 1470 ax-i5r 1471 ax-ext 2067 |
This theorem depends on definitions: df-bi 115 df-tru 1290 df-nf 1393 df-sb 1690 df-clab 2072 df-cleq 2078 df-clel 2081 df-nfc 2214 df-v 2616 df-in 2992 |
This theorem is referenced by: disjpr2 3483 rint0 3704 riin0 3778 xpriindim 4535 riinint 4655 reseq2 4669 csbresg 4677 resindm 4714 isoselem 5541 |
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