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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwnexch3and | Structured version Visualization version GIF version |
Description: Deduction form of btwnexch3 32716. (Contributed by Scott Fenton, 13-Oct-2013.) |
Ref | Expression |
---|---|
btwnexch3and.1 | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
btwnexch3and.2 | ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) |
btwnexch3and.3 | ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) |
btwnexch3and.4 | ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) |
btwnexch3and.5 | ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) |
btwnexch3and.6 | ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐶〉) |
btwnexch3and.7 | ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐴, 𝐷〉) |
Ref | Expression |
---|---|
btwnexch3and | ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐵, 𝐷〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | btwnexch3and.6 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐵 Btwn 〈𝐴, 𝐶〉) | |
2 | btwnexch3and.7 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐴, 𝐷〉) | |
3 | btwnexch3and.1 | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | btwnexch3and.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ (𝔼‘𝑁)) | |
5 | btwnexch3and.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ (𝔼‘𝑁)) | |
6 | btwnexch3and.4 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ (𝔼‘𝑁)) | |
7 | btwnexch3and.5 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝔼‘𝑁)) | |
8 | btwnexch3 32716 | . . . 4 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐶 Btwn 〈𝐵, 𝐷〉)) | |
9 | 3, 4, 5, 6, 7, 8 | syl122anc 1447 | . . 3 ⊢ (𝜑 → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐶 Btwn 〈𝐵, 𝐷〉)) |
10 | 9 | adantr 474 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐶 Btwn 〈𝐵, 𝐷〉)) |
11 | 1, 2, 10 | mp2and 689 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐵, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∈ wcel 2107 〈cop 4404 class class class wbr 4886 ‘cfv 6135 ℕcn 11374 𝔼cee 26237 Btwn cbtwn 26238 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-oi 8704 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-n0 11643 df-z 11729 df-uz 11993 df-rp 12138 df-ico 12493 df-icc 12494 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-hash 13436 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-clim 14627 df-sum 14825 df-ee 26240 df-btwn 26241 df-cgr 26242 |
This theorem is referenced by: cgrxfr 32751 btwnconn1lem1 32783 btwnconn1lem2 32784 btwnconn1lem12 32794 btwnconn3 32799 btwnoutside 32821 broutsideof3 32822 lineunray 32843 lineelsb2 32844 |
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