mmtheorems9 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 801-900 - Page 9 of 123

Type	Label	Description
Statement
Theorem	3simp2d 801	Deduce a conjunct from a triple conjunction.
|- (ph -> (ps /\ ch /\ th))   =>   |- (ph -> ch)
Theorem	3simp3d 802	Deduce a conjunct from a triple conjunction.
|- (ph -> (ps /\ ch /\ th))   =>   |- (ph -> th)
Theorem	3adant1 803	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- ((th /\ ph /\ ps) -> ch)
Theorem	3adant2 804	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- ((ph /\ th /\ ps) -> ch)
Theorem	3adant3 805	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- ((ph /\ ps /\ th) -> ch)
Theorem	3ad2ant1 806	Deduction adding conjuncts to an antecedent.
|- (ph -> ch)   =>   |- ((ph /\ ps /\ th) -> ch)
Theorem	3ad2ant2 807	Deduction adding conjuncts to an antecedent.
|- (ph -> ch)   =>   |- ((ps /\ ph /\ th) -> ch)
Theorem	3ad2ant3 808	Deduction adding conjuncts to an antecedent.
|- (ph -> ch)   =>   |- ((ps /\ th /\ ph) -> ch)
Theorem	3adantl1 809	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ta /\ ph /\ ps) /\ ch) -> th)
Theorem	3adantl2 810	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ta /\ ps) /\ ch) -> th)
Theorem	3adantl3 811	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ps /\ ta) /\ ch) -> th)
Theorem	3adantr1 812	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ta /\ ps /\ ch)) -> th)
Theorem	3adantr2 813	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3adantr3 814	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3ad2antl1 815	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ph /\ ps /\ ta) /\ ch) -> th)
Theorem	3ad2antl2 816	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ps /\ ph /\ ta) /\ ch) -> th)
Theorem	3ad2antl3 817	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ps /\ ta /\ ph) /\ ch) -> th)
Theorem	3ad2antr1 818	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ch /\ ps /\ ta)) -> th)
Theorem	3ad2antr2 819	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3ad2antr3 820	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3mix1 821	Introduction in triple disjunction.
|- (ph -> (ph \/ ps \/ ch))
Theorem	3mix2 822	Introduction in triple disjunction.
|- (ph -> (ps \/ ph \/ ch))
Theorem	3mix3 823	Introduction in triple disjunction.
|- (ph -> (ps \/ ch \/ ph))
Theorem	3pm3.2i 824	Infer conjunction of premises.
|- ph   &   |- ps   &   |- ch   =>   |- (ph /\ ps /\ ch)
Theorem	3jca 825	Join consequents with conjunction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   =>   |- (ph -> (ps /\ ch /\ th))
Theorem	3jcad 826	Deduction conjoining the consequents of three implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps -> th))   &   |- (ph -> (ps -> ta))   =>   |- (ph -> (ps -> (ch /\ th /\ ta)))
Theorem	3anim123i 827	Join antecedents and consequents with conjunction.
|- (ph -> ps)   &   |- (ch -> th)   &   |- (ta -> et)   =>   |- ((ph /\ ch /\ ta) -> (ps /\ th /\ et))
Theorem	3anbi123i 828	Join 3 biconditionals with conjunction.
|- (ph <-> ps)   &   |- (ch <-> th)   &   |- (ta <-> et)   =>   |- ((ph /\ ch /\ ta) <-> (ps /\ th /\ et))
Theorem	3orbi123i 829	Join 3 biconditionals with disjunction.
|- (ph <-> ps)   &   |- (ch <-> th)   &   |- (ta <-> et)   =>   |- ((ph \/ ch \/ ta) <-> (ps \/ th \/ et))
Theorem	3anbi1i 830	Inference adding two conjuncts to each side of a biconditional.
|- (ph <-> ps)   =>   |- ((ph /\ ch /\ th) <-> (ps /\ ch /\ th))
Theorem	3anbi2i 831	Inference adding two conjuncts to each side of a biconditional.
|- (ph <-> ps)   =>   |- ((ch /\ ph /\ th) <-> (ch /\ ps /\ th))
Theorem	3anbi3i 832	Inference adding two conjuncts to each side of a biconditional.
|- (ph <-> ps)   =>   |- ((ch /\ th /\ ph) <-> (ch /\ th /\ ps))
Theorem	3imp 833	Importation inference.
|- (ph -> (ps -> (ch -> th)))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impa 834	Importation from double to triple conjunction.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impb 835	Importation from double to triple conjunction.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impia 836	Importation to triple conjunction.
|- ((ph /\ ps) -> (ch -> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impib 837	Importation to triple conjunction.
|- (ph -> ((ps /\ ch) -> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3exp 838	Exportation inference.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> (ps -> (ch -> th)))
Theorem	3expa 839	Exportation from triple to double conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (((ph /\ ps) /\ ch) -> th)
Theorem	3expb 840	Exportation from triple to double conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	3expia 841	Exportation from triple conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> (ch -> th))
Theorem	3expib 842	Exportation from triple conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> ((ps /\ ch) -> th))
Theorem	3com12 843	Commutation in antecedent. Swap 1st and 3rd.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ph /\ ch) -> th)
Theorem	3com13 844	Commutation in antecedent. Swap 1st and 3rd.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ch /\ ps /\ ph) -> th)
Theorem	3com23 845	Commutation in antecedent. Swap 2nd and 3rd.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ch /\ ps) -> th)
Theorem	3coml 846	Commutation in antecedent. Rotate left.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ch /\ ph) -> th)
Theorem	3comr 847	Commutation in antecedent. Rotate right.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ch /\ ph /\ ps) -> th)
Theorem	3adant3r1 848	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ta /\ ps /\ ch)) -> th)
Theorem	3adant3r2 849	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3adant3r3 850	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3an1rs 851	Swap conjuncts.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ps /\ th) /\ ch) -> ta)
Theorem	3imp1 852	Importation to left triple conjunction.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (((ph /\ ps /\ ch) /\ th) -> ta)
Theorem	3impd 853	Importation deduction for triple conjunction.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (ph -> ((ps /\ ch /\ th) -> ta))
Theorem	3imp2 854	Importation to right triple conjunction.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- ((ph /\ (ps /\ ch /\ th)) -> ta)
Theorem	3exp1 855	Exportation from left triple conjunction.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	3expd 856	Exportation deduction for triple conjunction.
|- (ph -> ((ps /\ ch /\ th) -> ta))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	3exp2 857	Exportation from right triple conjunction.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	3adant1l 858	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (((ta /\ ph) /\ ps /\ ch) -> th)
Theorem	3adant1r 859	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (((ph /\ ta) /\ ps /\ ch) -> th)
Theorem	3adant2l 860	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ta /\ ps) /\ ch) -> th)
Theorem	3adant2r 861	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta) /\ ch) -> th)
Theorem	3adant3l 862	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps /\ (ta /\ ch)) -> th)
Theorem	3adant3r 863	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps /\ (ch /\ ta)) -> th)
Theorem	syl3anc 864	A syllogism inference combined with contraction.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ph)   &   |- (ta -> ps)   &   |- (ta -> ch)   =>   |- (ta -> th)
Theorem	syl3an1 865	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ph)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syl3an2 866	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ps)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3an3 867	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ch)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syl3an1b 868	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ph)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syl3an2b 869	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ps)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3an3b 870	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ch)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syl3an1br 871	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ph <-> ta)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syl3an2br 872	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ps <-> ta)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3an3br 873	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ch <-> ta)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syl3an 874	A triple syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta -> ph)   &   |- (et -> ps)   &   |- (ze -> ch)   =>   |- ((ta /\ et /\ ze) -> th)
Theorem	syl3anb 875	A triple syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- (ta <-> ph)   &   |- (et <-> ps)   &   |- (ze <-> ch)   =>   |- ((ta /\ et /\ ze) -> th)
Theorem	syld3an3 876	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ph /\ ps /\ ta) -> ch)   =>   |- ((ph /\ ps /\ ta) -> th)
Theorem	syld3an1 877	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ta /\ ps /\ ch) -> ph)   =>   |- ((ta /\ ps /\ ch) -> th)
Theorem	syld3an2 878	A syllogism inference.
|- ((ph /\ ps /\ ch) -> th)   &   |- ((ph /\ ta /\ ch) -> ps)   =>   |- ((ph /\ ta /\ ch) -> th)
Theorem	syl3anl1 879	A syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ph)   =>   |- (((et /\ ps /\ ch) /\ th) -> ta)
Theorem	syl3anl2 880	A syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ps)   =>   |- (((ph /\ et /\ ch) /\ th) -> ta)
Theorem	syl3anl3 881	A syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ch)   =>   |- (((ph /\ ps /\ et) /\ th) -> ta)
Theorem	syl3anl 882	A triple syllogism inference.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   &   |- (et -> ph)   &   |- (ze -> ps)   &   |- (si -> ch)   =>   |- (((et /\ ze /\ si) /\ th) -> ta)
Theorem	syl3anr1 883	A syllogism inference.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   &   |- (et -> ps)   =>   |- ((ph /\ (et /\ ch /\ th)) -> ta)
Theorem	syl3anr2 884	A syllogism inference.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   &   |- (et -> ch)   =>   |- ((ph /\ (ps /\ et /\ th)) -> ta)
Theorem	syl3anr3 885	A syllogism inference.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   &   |- (et -> th)   =>   |- ((ph /\ (ps /\ ch /\ et)) -> ta)
Theorem	3impdi 886	Importation inference (undistribute conjunction).
|- (((ph /\ ps) /\ (ph /\ ch)) -> th)   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impdir 887	Importation inference (undistribute conjunction).
|- (((ph /\ ps) /\ (ch /\ ps)) -> th)   =>   |- ((ph /\ ch /\ ps) -> th)
Theorem	3anidm12 888	Inference from idempotent law for conjunction.
|- ((ph /\ ph /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	3anidm13 889	Inference from idempotent law for conjunction.
|- ((ph /\ ps /\ ph) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	3anidm23 890	Inference from idempotent law for conjunction.
|- ((ph /\ ps /\ ps) -> ch)   =>   |- ((ph /\ ps) -> ch)
Theorem	3ori 891	Infer implication from triple disjunction.
|- (ph \/ ps \/ ch)   =>   |- ((-. ph /\ -. ps) -> ch)
Theorem	3jao 892	Disjunction of 3 antecedents.
|- (((ph -> ps) /\ (ch -> ps) /\ (th -> ps)) -> ((ph \/ ch \/ th) -> ps))
Theorem	3jaoi 893	Disjunction of 3 antecedents (inference).
|- (ph -> ps)   &   |- (ch -> ps)   &   |- (th -> ps)   =>   |- ((ph \/ ch \/ th) -> ps)
Theorem	3jaod 894	Disjunction of 3 antecedents (deduction).
|- (ph -> (ps -> ch))   &   |- (ph -> (th -> ch))   &   |- (ph -> (ta -> ch))   =>   |- (ph -> ((ps \/ th \/ ta) -> ch))
Theorem	3jaoian 895	Disjunction of 3 antecedents (inference).
|- ((ph /\ ps) -> ch)   &   |- ((th /\ ps) -> ch)   &   |- ((ta /\ ps) -> ch)   =>   |- (((ph \/ th \/ ta) /\ ps) -> ch)
Theorem	3jaodan 896	Disjunction of 3 antecedents (deduction).
|- ((ph /\ ps) -> ch)   &   |- ((ph /\ th) -> ch)   &   |- ((ph /\ ta) -> ch)   =>   |- ((ph /\ (ps \/ th \/ ta)) -> ch)
Theorem	syl3an9b 897	Nested syllogism inference conjoining 3 dissimilar antecedents.
|- (ph -> (ps <-> ch))   &   |- (th -> (ch <-> ta))   &   |- (et -> (ta <-> ze))   =>   |- ((ph /\ th /\ et) -> (ps <-> ze))
Theorem	3orbi123d 898	Deduction joining 3 equivalences to form equivalence of disjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   &   |- (ph -> (et <-> ze))   =>   |- (ph -> ((ps \/ th \/ et) <-> (ch \/ ta \/ ze)))
Theorem	3anbi123d 899	Deduction joining 3 equivalences to form equivalence of conjunctions.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   &   |- (ph -> (et <-> ze))   =>   |- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ ze)))
Theorem	3anbi12d 900	Deduction conjoining and adding a conjunct to equivalences.
|- (ph -> (ps <-> ch))   &   |- (ph -> (th <-> ta))   =>   |- (ph -> ((ps /\ th /\ et) <-> (ch /\ ta /\ et)))