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Mirrors > Home > MPE Home > Th. List > Mathboxes > btwnexch3and | Structured version Visualization version GIF version |
Description: Deduction form of btwnexch3 35986. (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 35986 | . . . 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 1379 | . . 3 ⊢ (𝜑 → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐶 Btwn 〈𝐵, 𝐷〉)) |
10 | 9 | adantr 480 | . 2 ⊢ ((𝜑 ∧ 𝜓) → ((𝐵 Btwn 〈𝐴, 𝐶〉 ∧ 𝐶 Btwn 〈𝐴, 𝐷〉) → 𝐶 Btwn 〈𝐵, 𝐷〉)) |
11 | 1, 2, 10 | mp2and 698 | 1 ⊢ ((𝜑 ∧ 𝜓) → 𝐶 Btwn 〈𝐵, 𝐷〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 〈cop 4654 class class class wbr 5166 ‘cfv 6575 ℕcn 12295 𝔼cee 28923 Btwn cbtwn 28924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7772 ax-inf2 9712 ax-cnex 11242 ax-resscn 11243 ax-1cn 11244 ax-icn 11245 ax-addcl 11246 ax-addrcl 11247 ax-mulcl 11248 ax-mulrcl 11249 ax-mulcom 11250 ax-addass 11251 ax-mulass 11252 ax-distr 11253 ax-i2m1 11254 ax-1ne0 11255 ax-1rid 11256 ax-rnegex 11257 ax-rrecex 11258 ax-cnre 11259 ax-pre-lttri 11260 ax-pre-lttrn 11261 ax-pre-ltadd 11262 ax-pre-mulgt0 11263 ax-pre-sup 11264 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6334 df-ord 6400 df-on 6401 df-lim 6402 df-suc 6403 df-iota 6527 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-fv 6583 df-isom 6584 df-riota 7406 df-ov 7453 df-oprab 7454 df-mpo 7455 df-om 7906 df-1st 8032 df-2nd 8033 df-frecs 8324 df-wrecs 8355 df-recs 8429 df-rdg 8468 df-1o 8524 df-er 8765 df-map 8888 df-en 9006 df-dom 9007 df-sdom 9008 df-fin 9009 df-sup 9513 df-oi 9581 df-card 10010 df-pnf 11328 df-mnf 11329 df-xr 11330 df-ltxr 11331 df-le 11332 df-sub 11524 df-neg 11525 df-div 11950 df-nn 12296 df-2 12358 df-3 12359 df-n0 12556 df-z 12642 df-uz 12906 df-rp 13060 df-ico 13415 df-icc 13416 df-fz 13570 df-fzo 13714 df-seq 14055 df-exp 14115 df-hash 14382 df-cj 15150 df-re 15151 df-im 15152 df-sqrt 15286 df-abs 15287 df-clim 15536 df-sum 15737 df-ee 28926 df-btwn 28927 df-cgr 28928 |
This theorem is referenced by: cgrxfr 36021 btwnconn1lem1 36053 btwnconn1lem2 36054 btwnconn1lem12 36064 btwnconn3 36069 btwnoutside 36091 broutsideof3 36092 lineunray 36113 lineelsb2 36114 |
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