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Mirrors > Home > NFE Home > Th. List > ineq12 | GIF version |
Description: Equality theorem for intersection of two classes. (Contributed by NM, 8-May-1994.) |
Ref | Expression |
---|---|
ineq12 | ⊢ ((A = B ∧ C = D) → (A ∩ C) = (B ∩ D)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1 3450 | . 2 ⊢ (A = B → (A ∩ C) = (B ∩ C)) | |
2 | ineq2 3451 | . 2 ⊢ (C = D → (B ∩ C) = (B ∩ D)) | |
3 | 1, 2 | sylan9eq 2405 | 1 ⊢ ((A = B ∧ C = D) → (A ∩ C) = (B ∩ D)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 358 = wceq 1642 ∩ cin 3208 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-in 3213 |
This theorem is referenced by: ineq12i 3455 ineq12d 3458 ineqan12d 3459 fnun 5189 fvun1 5379 fntxp 5804 endisj 6051 ncdisjeq 6148 letc 6231 |
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