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Mirrors > Home > ILE Home > Th. List > mpancom | GIF version |
Description: An inference based on modus ponens with commutation of antecedents. (Contributed by NM, 28-Oct-2003.) (Proof shortened by Wolf Lammen, 7-Apr-2013.) |
Ref | Expression |
---|---|
mpancom.1 | ⊢ (𝜓 → 𝜑) |
mpancom.2 | ⊢ ((𝜑 ∧ 𝜓) → 𝜒) |
Ref | Expression |
---|---|
mpancom | ⊢ (𝜓 → 𝜒) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpancom.1 | . 2 ⊢ (𝜓 → 𝜑) | |
2 | id 19 | . 2 ⊢ (𝜓 → 𝜓) | |
3 | mpancom.2 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜒) | |
4 | 1, 2, 3 | syl2anc 411 | 1 ⊢ (𝜓 → 𝜒) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia3 108 |
This theorem is referenced by: mpan 424 spesbc 3050 onsucelsucr 4509 sucunielr 4511 ordsuc 4564 peano2b 4616 xpiindim 4766 fvelrnb 5565 fliftcnv 5798 riotaprop 5856 unielxp 6177 dmtpos 6259 tpossym 6279 ercnv 6558 cnvct 6811 php5dom 6865 3xpfi 6932 recrecnq 7395 1idpr 7593 eqlei2 8054 lem1 8806 eluzfz1 10033 fzpred 10072 uznfz 10105 fz0fzdiffz0 10132 fzctr 10135 flid 10286 flqeqceilz 10320 faclbnd3 10725 bcn1 10740 isfinite4im 10774 leabs 11085 gcd0id 11982 lcmgcdlem 12079 dvdsnprmd 12127 pcprod 12346 fldivp1 12348 intopsn 12791 mgm1 12794 sgrp1 12821 mnd1 12852 mnd1id 12853 grp1 12981 grp1inv 12982 eqger 13088 eqgid 13090 ring1 13241 ringressid 13243 subrgsubm 13360 cncrng 13502 eltpsg 13579 tg1 13598 cldval 13638 cldss 13644 cldopn 13646 psmetdmdm 13863 dvef 14227 relogef 14324 zabsle1 14439 bj-nn0suc0 14741 |
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