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Mirrors > Home > MPE Home > Th. List > ancld | Structured version Visualization version GIF version |
Description: Deduction conjoining antecedent to left of consequent in nested implication. (Contributed by NM, 15-Aug-1994.) (Proof shortened by Wolf Lammen, 1-Nov-2012.) |
Ref | Expression |
---|---|
ancld.1 | ⊢ (𝜑 → (𝜓 → 𝜒)) |
Ref | Expression |
---|---|
ancld | ⊢ (𝜑 → (𝜓 → (𝜓 ∧ 𝜒))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | idd 24 | . 2 ⊢ (𝜑 → (𝜓 → 𝜓)) | |
2 | ancld.1 | . 2 ⊢ (𝜑 → (𝜓 → 𝜒)) | |
3 | 1, 2 | jcad 512 | 1 ⊢ (𝜑 → (𝜓 → (𝜓 ∧ 𝜒))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 207 df-an 396 |
This theorem is referenced by: dfmoeu 2533 mopick2 2634 2eu6 2654 cgsexg 3523 cgsex2g 3524 cgsex4g 3525 cgsex4gOLD 3526 reximdva0 4360 difsn 4802 preq12b 4854 elinxp 6038 ssrnres 6199 ordtr2 6429 elunirn 7270 fnoprabg 7555 tz7.49 8483 omord 8604 ficard 10602 fpwwe2lem11 10678 1idpr 11066 xrsupsslem 13345 xrinfmsslem 13346 fzospliti 13727 sqrt2irr 16281 algcvga 16612 prmind2 16718 infpn2 16946 grpinveu 19004 1stcrest 23476 fgss2 23897 fgcl 23901 filufint 23943 metrest 24552 reconnlem2 24862 plydivex 26353 rtprmirr 26817 ftalem3 27132 chtub 27270 lgsqrmodndvds 27411 2sqlem10 27486 dchrisum0flb 27568 pntpbnd1 27644 nolesgn2o 27730 nosupbnd1lem4 27770 noinfbnd1lem4 27785 noetalem1 27800 clwwlkn1loopb 30071 2pthfrgrrn2 30311 grpoidinvlem3 30534 grpoinveu 30547 elim2ifim 32565 iocinif 32789 qsxpid 33369 tpr2rico 33872 bnj168 34722 ellcsrspsn 35625 dfon2lem8 35771 nn0prpwlem 36304 bj-opelidres 37143 difunieq 37356 voliunnfl 37650 dalem20 39675 elpaddn0 39782 cdleme25a 40335 cdleme29ex 40356 cdlemefr29exN 40384 dibglbN 41148 dihlsscpre 41216 lcfl7N 41483 mapdh9a 41771 mapdh9aOLDN 41772 hdmap11lem2 41824 eu6w 42662 sqrtcval 43630 ax6e2eq 44554 eliin2f 45043 clnbgr3stgrgrlic 47914 itschlc0xyqsol1 48615 mpbiran3d 48645 |
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