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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 3312 | . 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 1342 ∩ cin 3110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-v 2723 df-in 3117 |
This theorem is referenced by: disjpr2 3634 rint0 3857 riin0 3931 disji2 3969 xpriindim 4736 riinint 4859 reseq2 4873 csbresg 4881 resindm 4920 isoselem 5782 zfz1isolem1 10739 fsumm1 11343 ennnfonelemhf1o 12289 nninfdclemcl 12326 nninfdclemp1 12328 nninfdc 12331 restval 12504 basis1 12592 baspartn 12595 eltg 12599 tgdom 12619 ntrval 12657 resttopon2 12725 restopnb 12728 qtopbasss 13068 |
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