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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 3048 onsucelsucr 4507 sucunielr 4509 ordsuc 4562 peano2b 4614 xpiindim 4764 fvelrnb 5563 fliftcnv 5795 riotaprop 5853 unielxp 6174 dmtpos 6256 tpossym 6276 ercnv 6555 cnvct 6808 php5dom 6862 3xpfi 6929 recrecnq 7392 1idpr 7590 eqlei2 8051 lem1 8803 eluzfz1 10030 fzpred 10069 uznfz 10102 fz0fzdiffz0 10129 fzctr 10132 flid 10283 flqeqceilz 10317 faclbnd3 10722 bcn1 10737 isfinite4im 10771 leabs 11082 gcd0id 11979 lcmgcdlem 12076 dvdsnprmd 12124 pcprod 12343 fldivp1 12345 intopsn 12785 mgm1 12788 sgrp1 12815 mnd1 12846 mnd1id 12847 grp1 12975 grp1inv 12976 eqger 13081 eqgid 13083 ring1 13234 ringressid 13236 subrgsubm 13353 cncrng 13433 eltpsg 13510 tg1 13529 cldval 13569 cldss 13575 cldopn 13577 psmetdmdm 13794 dvef 14158 relogef 14255 zabsle1 14370 bj-nn0suc0 14672 |
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