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Mirrors > Home > ILE Home > Th. List > df-3an | GIF version |
Description: Define conjunction ('and') of 3 wff.s. Definition *4.34 of [WhiteheadRussell] p. 118. This abbreviation reduces the number of parentheses and emphasizes that the order of bracketing is not important by virtue of the associative law anass 399. (Contributed by NM, 8-Apr-1994.) |
Ref | Expression |
---|---|
df-3an | ⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wph | . . 3 wff 𝜑 | |
2 | wps | . . 3 wff 𝜓 | |
3 | wch | . . 3 wff 𝜒 | |
4 | 1, 2, 3 | w3a 963 | . 2 wff (𝜑 ∧ 𝜓 ∧ 𝜒) |
5 | 1, 2 | wa 103 | . . 3 wff (𝜑 ∧ 𝜓) |
6 | 5, 3 | wa 103 | . 2 wff ((𝜑 ∧ 𝜓) ∧ 𝜒) |
7 | 4, 6 | wb 104 | 1 wff ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ((𝜑 ∧ 𝜓) ∧ 𝜒)) |
Colors of variables: wff set class |
This definition is referenced by: 3anass 967 3anrot 968 3ancoma 970 3anan32 974 3ioran 978 3simpa 979 3pm3.2i 1160 pm3.2an3 1161 3jca 1162 3anbi123i 1171 3imp 1176 3anbi123d 1291 3anim123d 1298 an6 1300 19.26-3an 1460 hb3an 1530 nf3an 1546 nf3and 1549 eeeanv 1906 sb3an 1932 mopick2 2083 r19.26-3 2565 3reeanv 2604 ceqsex3v 2731 ceqsex4v 2732 ceqsex8v 2734 sbc3an 2974 elin3 3272 rexdifpr 3560 raltpg 3584 tpss 3693 dfopg 3711 opeq1 3713 opeq2 3714 opm 4164 otth2 4171 poirr 4237 po3nr 4240 wepo 4289 wetrep 4290 rabxp 4584 brinxp2 4614 brinxp 4615 sotri2 4944 sotri3 4945 f1orn 5385 dff1o6 5685 isosolem 5733 oprabid 5811 caovimo 5972 elovmpo 5979 dfxp3 6100 nnaord 6413 fiintim 6825 prmuloc 7398 ltrelxr 7849 rexuz2 9403 ltxr 9592 elixx3g 9714 elioo4g 9747 elioopnf 9780 elioomnf 9781 elicopnf 9782 elxrge0 9791 divelunit 9815 elfz2 9828 elfzuzb 9831 uzsplit 9903 fznn0 9924 elfzmlbp 9940 elfzo2 9958 fzolb2 9962 fzouzsplit 9987 ssfzo12bi 10033 fzind2 10047 abs2dif 10910 sumeq2 11160 divalgb 11658 divgcdz 11696 rplpwr 11751 cncongr1 11820 isbasis3g 12252 lmfval 12400 lmbr 12421 lmbr2 12422 xmeterval 12643 xmeter 12644 cnbl0 12742 cnblcld 12743 limcrcl 12835 bd3an 13199 |
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