mmtheorems9 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 801-900 - Page 9 of 107

Type	Label	Description
Statement
Theorem	3ad2ant3 801	Deduction adding conjuncts to an antecedent.
|- (ph -> ch)   =>   |- ((ps /\ th /\ ph) -> ch)
Theorem	3adantl1 802	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ta /\ ph /\ ps) /\ ch) -> th)
Theorem	3adantl2 803	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ta /\ ps) /\ ch) -> th)
Theorem	3adantl3 804	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ ps /\ ta) /\ ch) -> th)
Theorem	3adantr1 805	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ta /\ ps /\ ch)) -> th)
Theorem	3adantr2 806	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3adantr3 807	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3ad2antl1 808	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ph /\ ps /\ ta) /\ ch) -> th)
Theorem	3ad2antl2 809	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ps /\ ph /\ ta) /\ ch) -> th)
Theorem	3ad2antl3 810	Deduction adding conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- (((ps /\ ta /\ ph) /\ ch) -> th)
Theorem	3ad2antr1 811	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ch /\ ps /\ ta)) -> th)
Theorem	3ad2antr2 812	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3ad2antr3 813	Deduction adding a conjuncts to antecedent.
|- ((ph /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3mix1 814	Introduction in triple disjunction.
|- (ph -> (ph \/ ps \/ ch))
Theorem	3mix2 815	Introduction in triple disjunction.
|- (ph -> (ps \/ ph \/ ch))
Theorem	3mix3 816	Introduction in triple disjunction.
|- (ph -> (ps \/ ch \/ ph))
Theorem	3pm3.2i 817	Infer conjunction of premises.
|- ph   &   |- ps   &   |- ch   =>   |- (ph /\ ps /\ ch)
Theorem	3jca 818	Join consequents with conjunction.
|- (ph -> ps)   &   |- (ph -> ch)   &   |- (ph -> th)   =>   |- (ph -> (ps /\ ch /\ th))
Theorem	3jcad 819	Deduction conjoining the consequents of three implications.
|- (ph -> (ps -> ch))   &   |- (ph -> (ps -> th))   &   |- (ph -> (ps -> ta))   =>   |- (ph -> (ps -> (ch /\ th /\ ta)))
Theorem	3anim123i 820	Join antecedents and consequents with conjunction.
|- (ph -> ps)   &   |- (ch -> th)   &   |- (ta -> et)   =>   |- ((ph /\ ch /\ ta) -> (ps /\ th /\ et))
Theorem	3anbi123i 821	Join 3 biconditionals with conjunction.
|- (ph <-> ps)   &   |- (ch <-> th)   &   |- (ta <-> et)   =>   |- ((ph /\ ch /\ ta) <-> (ps /\ th /\ et))
Theorem	3orbi123i 822	Join 3 biconditionals with disjunction.
|- (ph <-> ps)   &   |- (ch <-> th)   &   |- (ta <-> et)   =>   |- ((ph \/ ch \/ ta) <-> (ps \/ th \/ et))
Theorem	3anbi1i 823	Inference adding two conjuncts to each side of a biconditional.
|- (ph <-> ps)   =>   |- ((ph /\ ch /\ th) <-> (ps /\ ch /\ th))
Theorem	3anbi2i 824	Inference adding two conjuncts to each side of a biconditional.
|- (ph <-> ps)   =>   |- ((ch /\ ph /\ th) <-> (ch /\ ps /\ th))
Theorem	3anbi3i 825	Inference adding two conjuncts to each side of a biconditional.
|- (ph <-> ps)   =>   |- ((ch /\ th /\ ph) <-> (ch /\ th /\ ps))
Theorem	3imp 826	Importation inference.
|- (ph -> (ps -> (ch -> th)))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impa 827	Importation from double to triple conjunction.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impb 828	Importation from double to triple conjunction.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impia 829	Importation to triple conjunction.
|- ((ph /\ ps) -> (ch -> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3impib 830	Importation to triple conjunction.
|- (ph -> ((ps /\ ch) -> th))   =>   |- ((ph /\ ps /\ ch) -> th)
Theorem	3exp 831	Exportation inference.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> (ps -> (ch -> th)))
Theorem	3expa 832	Exportation from triple to double conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (((ph /\ ps) /\ ch) -> th)
Theorem	3expb 833	Exportation from triple to double conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	3expia 834	Exportation from triple conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps) -> (ch -> th))
Theorem	3expib 835	Exportation from triple conjunction.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (ph -> ((ps /\ ch) -> th))
Theorem	3com12 836	Commutation in antecedent. Swap 1st and 3rd.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ph /\ ch) -> th)
Theorem	3com13 837	Commutation in antecedent. Swap 1st and 3rd.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ch /\ ps /\ ph) -> th)
Theorem	3com23 838	Commutation in antecedent. Swap 2nd and 3rd.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ch /\ ps) -> th)
Theorem	3coml 839	Commutation in antecedent. Rotate left.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ps /\ ch /\ ph) -> th)
Theorem	3comr 840	Commutation in antecedent. Rotate right.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ch /\ ph /\ ps) -> th)
Theorem	3adant3r1 841	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ta /\ ps /\ ch)) -> th)
Theorem	3adant3r2 842	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta /\ ch)) -> th)
Theorem	3adant3r3 843	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ch /\ ta)) -> th)
Theorem	3an1rs 844	Swap conjuncts.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (((ph /\ ps /\ th) /\ ch) -> ta)
Theorem	3imp1 845	Importation to left triple conjunction.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (((ph /\ ps /\ ch) /\ th) -> ta)
Theorem	3impd 846	Importation deduction for triple conjunction.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (ph -> ((ps /\ ch /\ th) -> ta))
Theorem	3imp2 847	Importation to right triple conjunction.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- ((ph /\ (ps /\ ch /\ th)) -> ta)
Theorem	3exp1 848	Exportation from left triple conjunction.
|- (((ph /\ ps /\ ch) /\ th) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	3expd 849	Exportation deduction for triple conjunction.
|- (ph -> ((ps /\ ch /\ th) -> ta))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	3exp2 850	Exportation from right triple conjunction.
|- ((ph /\ (ps /\ ch /\ th)) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	3adant1l 851	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (((ta /\ ph) /\ ps /\ ch) -> th)
Theorem	3adant1r 852	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- (((ph /\ ta) /\ ps /\ ch) -> th)
Theorem	3adant2l 853	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ta /\ ps) /\ ch) -> th)
Theorem	3adant2r 854	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ (ps /\ ta) /\ ch) -> th)
Theorem	3adant3l 855	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\ ch) -> th)   =>   |- ((ph /\ ps /\ (ta /\ ch)) -> th)
Theorem	3adant3r 856	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps /\