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Mirrors > Home > ILE Home > Th. List > ineq2i | GIF version |
Description: Equality inference for intersection of two classes. (Contributed by NM, 26-Dec-1993.) |
Ref | Expression |
---|---|
ineq1i.1 | ⊢ 𝐴 = 𝐵 |
Ref | Expression |
---|---|
ineq2i | ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | ineq2 3198 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵)) | |
3 | 1, 2 | ax-mp 7 | 1 ⊢ (𝐶 ∩ 𝐴) = (𝐶 ∩ 𝐵) |
Colors of variables: wff set class |
Syntax hints: = wceq 1290 ∩ cin 3001 |
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-io 666 ax-5 1382 ax-7 1383 ax-gen 1384 ax-ie1 1428 ax-ie2 1429 ax-8 1441 ax-10 1442 ax-11 1443 ax-i12 1444 ax-bndl 1445 ax-4 1446 ax-17 1465 ax-i9 1469 ax-ial 1473 ax-i5r 1474 ax-ext 2071 |
This theorem depends on definitions: df-bi 116 df-tru 1293 df-nf 1396 df-sb 1694 df-clab 2076 df-cleq 2082 df-clel 2085 df-nfc 2218 df-v 2624 df-in 3008 |
This theorem is referenced by: in4 3219 inindir 3221 indif2 3246 difun1 3262 dfrab3ss 3280 dfif3 3412 intunsn 3734 rint0 3735 riin0 3809 res0 4732 resres 4740 resundi 4741 resindi 4743 inres 4745 resiun2 4748 resopab 4771 dfse2 4820 dminxp 4890 imainrect 4891 resdmres 4937 funimacnv 5105 unfiin 6692 sbthlemi5 6726 dmaddpi 6947 dmmulpi 6948 fsumiun 10934 |
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