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Mirrors > Home > NFE Home > Th. List > ineq12i | GIF version |
Description: Equality inference for intersection of two classes. (Contributed by NM, 24-Jun-2004.) (Proof shortened by Eric Schmidt, 26-Jan-2007.) |
Ref | Expression |
---|---|
ineq1i.1 | ⊢ A = B |
ineq12i.2 | ⊢ C = D |
Ref | Expression |
---|---|
ineq12i | ⊢ (A ∩ C) = (B ∩ D) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1i.1 | . 2 ⊢ A = B | |
2 | ineq12i.2 | . 2 ⊢ C = D | |
3 | ineq12 3452 | . 2 ⊢ ((A = B ∧ C = D) → (A ∩ C) = (B ∩ D)) | |
4 | 1, 2, 3 | mp2an 653 | 1 ⊢ (A ∩ C) = (B ∩ D) |
Colors of variables: wff setvar class |
Syntax hints: = 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: undir 3504 difundi 3507 difindir 3510 inrab 3527 inrab2 3528 iinun 3548 dfif4 3673 dfif5 3674 evenodddisj 4516 inxp 4863 resindi 4983 resindir 4984 rnin 5037 cnvpprod 5841 pmex 6005 sbthlem1 6203 nmembers1lem1 6268 |
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