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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 3237 | . 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 1314 ∩ cin 3036 |
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 681 ax-5 1406 ax-7 1407 ax-gen 1408 ax-ie1 1452 ax-ie2 1453 ax-8 1465 ax-10 1466 ax-11 1467 ax-i12 1468 ax-bndl 1469 ax-4 1470 ax-17 1489 ax-i9 1493 ax-ial 1497 ax-i5r 1498 ax-ext 2097 |
This theorem depends on definitions: df-bi 116 df-tru 1317 df-nf 1420 df-sb 1719 df-clab 2102 df-cleq 2108 df-clel 2111 df-nfc 2244 df-v 2659 df-in 3043 |
This theorem is referenced by: disjpr2 3553 rint0 3776 riin0 3850 disji2 3888 xpriindim 4637 riinint 4758 reseq2 4772 csbresg 4780 resindm 4819 isoselem 5675 zfz1isolem1 10476 fsumm1 11077 ennnfonelemhf1o 11771 restval 11969 basis1 12057 baspartn 12060 eltg 12064 tgdom 12084 ntrval 12122 resttopon2 12190 restopnb 12193 qtopbasss 12510 |
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