![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 3196 | . 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 2999 |
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 2622 df-in 3006 |
This theorem is referenced by: in4 3217 inindir 3219 indif2 3244 difun1 3260 dfrab3ss 3278 dfif3 3410 intunsn 3732 rint0 3733 riin0 3807 res0 4730 resres 4738 resundi 4739 resindi 4741 inres 4743 resiun2 4746 resopab 4769 dfse2 4818 dminxp 4888 imainrect 4889 resdmres 4935 funimacnv 5103 unfiin 6690 sbthlemi5 6724 dmaddpi 6945 dmmulpi 6946 fsumiun 10932 |
Copyright terms: Public domain | W3C validator |