mmtheorems4 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 301-400 - Page 4 of 123

Type	Label	Description
Statement
Theorem	anc2ri 301	Deduction conjoining antecedent to right of consequent in nested implication.
|- (ph -> (ps -> ch))   =>   |- (ph -> (ps -> (ch /\ ph)))
Theorem	anor 302	Conjunction in terms of disjunction (DeMorgan's law). Theorem *4.5 of [WhiteheadRussell] p. 120.
|- ((ph /\ ps) <-> -. (-. ph \/ -. ps))
Theorem	ianor 303	Negated conjunction in terms of disjunction (DeMorgan's law). Theorem *4.51 of [WhiteheadRussell] p. 120.
|- (-. (ph /\ ps) <-> (-. ph \/ -. ps))
Theorem	ioran 304	Negated disjunction in terms of conjunction (DeMorgan's law). Compare Theorem *4.56 of [WhiteheadRussell] p. 120.
|- (-. (ph \/ ps) <-> (-. ph /\ -. ps))
Theorem	pm4.52 305	Theorem *4.52 of [WhiteheadRussell] p. 120.
|- ((ph /\ -. ps) <-> -. (-. ph \/ ps))
Theorem	pm4.53 306	Theorem *4.53 of [WhiteheadRussell] p. 120.
|- (-. (ph /\ -. ps) <-> (-. ph \/ ps))
Theorem	pm4.54 307	Theorem *4.54 of [WhiteheadRussell] p. 120.
|- ((-. ph /\ ps) <-> -. (ph \/ -. ps))
Theorem	pm4.55 308	Theorem *4.55 of [WhiteheadRussell] p. 120.
|- (-. (-. ph /\ ps) <-> (ph \/ -. ps))
Theorem	pm4.56 309	Theorem *4.56 of [WhiteheadRussell] p. 120.
|- ((-. ph /\ -. ps) <-> -. (ph \/ ps))
Theorem	oran 310	Disjunction in terms of conjunction (DeMorgan's law). Compare Theorem *4.57 of [WhiteheadRussell] p. 120.
|- ((ph \/ ps) <-> -. (-. ph /\ -. ps))
Theorem	pm4.57 311	Theorem *4.57 of [WhiteheadRussell] p. 120.
|- (-. (-. ph /\ -. ps) <-> (ph \/ ps))
Theorem	pm3.1 312	Theorem *3.1 of [WhiteheadRussell] p. 111.
|- ((ph /\ ps) -> -. (-. ph \/ -. ps))
Theorem	pm3.11 313	Theorem *3.11 of [WhiteheadRussell] p. 111.
|- (-. (-. ph \/ -. ps) -> (ph /\ ps))
Theorem	pm3.12 314	Theorem *3.12 of [WhiteheadRussell] p. 111.
|- ((-. ph \/ -. ps) \/ (ph /\ ps))
Theorem	pm3.13 315	Theorem *3.13 of [WhiteheadRussell] p. 111.
|- (-. (ph /\ ps) -> (-. ph \/ -. ps))
Theorem	pm3.14 316	Theorem *3.14 of [WhiteheadRussell] p. 111.
|- ((-. ph \/ -. ps) -> -. (ph /\ ps))
Theorem	pm3.26 317	Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112.
|- ((ph /\ ps) -> ph)
Theorem	pm3.26i 318	Inference eliminating a conjunct.
|- (ph /\ ps)   =>   |- ph
Theorem	pm3.26d 319	Deduction eliminating a conjunct.
|- (ph -> (ps /\ ch))   =>   |- (ph -> ps)
Theorem	pm3.26bi 320	Deduction eliminating a conjunct.
|- (ph <-> (ps /\ ch))   =>   |- (ph -> ps)
Theorem	pm3.27 321	Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112.
|- ((ph /\ ps) -> ps)
Theorem	pm3.27i 322	Inference eliminating a conjunct.
|- (ph /\ ps)   =>   |- ps
Theorem	pm3.27d 323	Deduction eliminating a conjunct.
|- (ph -> (ps /\ ch))   =>   |- (ph -> ch)
Theorem	pm3.27bi 324	Deduction eliminating a conjunct.
|- (ph <-> (ps /\ ch))   =>   |- (ph -> ch)
Theorem	pm3.41 325	Theorem *3.41 of [WhiteheadRussell] p. 113.
|- ((ph -> ch) -> ((ph /\ ps) -> ch))
Theorem	pm3.42 326	Theorem *3.42 of [WhiteheadRussell] p. 113.
|- ((ps -> ch) -> ((ph /\ ps) -> ch))
Theorem	anclb 327	Conjoin antecedent to left of consequent. Theorem *4.7 of [WhiteheadRussell] p. 120.
|- ((ph -> ps) <-> (ph -> (ph /\ ps)))
Theorem	ancrb 328	Conjoin antecedent to right of consequent.
|- ((ph -> ps) <-> (ph -> (ps /\ ph)))
Theorem	pm3.4 329	Conjunction implies implication. Theorem *3.4 of [WhiteheadRussell] p. 113.
|- ((ph /\ ps) -> (ph -> ps))
Theorem	pm4.45im 330	Conjunction with implication. Compare Theorem *4.45 of [WhiteheadRussell] p. 119.
|- (ph <-> (ph /\ (ps -> ph)))
Theorem	anim12i 331	Conjoin antecedents and consequents of two premises.
|- (ph -> ps)   &   |- (ch -> th)   =>   |- ((ph /\ ch) -> (ps /\ th))
Theorem	anim1i 332	Introduce conjunct to both sides of an implication.
|- (ph -> ps)   =>   |- ((ph /\ ch) -> (ps /\ ch))
Theorem	anim2i 333	Introduce conjunct to both sides of an implication.
|- (ph -> ps)   =>   |- ((ch /\ ph) -> (ch /\ ps))
Theorem	orim12i 334	Disjoin antecedents and consequents of two premises.
|- (ph -> ps)   &   |- (ch -> th)   =>   |- ((ph \/ ch) -> (ps \/ th))
Theorem	orim1i 335	Introduce disjunct to both sides of an implication.
|- (ph -> ps)   =>   |- ((ph \/ ch) -> (ps \/ ch))
Theorem	orim2i 336	Introduce disjunct to both sides of an implication.
|- (ph -> ps)   =>   |- ((ch \/ ph) -> (ch \/ ps))
Theorem	pm2.3 337	Theorem *2.3 of [WhiteheadRussell] p. 104.
|- ((ph \/ (ps \/ ch)) -> (ph \/ (ch \/ ps)))
Theorem	jao 338	Disjunction of antecedents. Compare Theorem *3.44 of [WhiteheadRussell] p. 113.
|- ((ph -> ps) -> ((ch -> ps) -> ((ph \/ ch) -> ps)))
Theorem	jaoi 339	Inference disjoining the antecedents of two implications.
|- (ph -> ps)   &   |- (ch -> ps)   =>   |- ((ph \/ ch) -> ps)
Theorem	pm2.41 340	Theorem *2.41 of [WhiteheadRussell] p. 106.
|- ((ps \/ (ph \/ ps)) -> (ph \/ ps))
Theorem	pm2.42 341	Theorem *2.42 of [WhiteheadRussell] p. 106.
|- ((-. ph \/ (ph -> ps)) -> (ph -> ps))
Theorem	pm2.4 342	Theorem *2.4 of [WhiteheadRussell] p. 106.
|- ((ph \/ (ph \/ ps)) -> (ph \/ ps))
Theorem	pm4.44 343	Theorem *4.44 of [WhiteheadRussell] p. 119.
|- (ph <-> (ph \/ (ph /\ ps)))
Theorem	pm5.63 344	Theorem *5.63 of [WhiteheadRussell] p. 125.
|- ((ph \/ ps) <-> (ph \/ (-. ph /\ ps)))
Theorem	impexp 345	Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell] p. 122.
|- (((ph /\ ps) -> ch) <-> (ph -> (ps -> ch)))
Theorem	pm3.3 346	Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112.
|- (((ph /\ ps) -> ch) -> (ph -> (ps -> ch)))
Theorem	pm3.31 347	Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112.
|- ((ph -> (ps -> ch)) -> ((ph /\ ps) -> ch))
Theorem	imp 348	Importation inference. (The proof was shortened by Eric Schmidt, 22-Dec-2006.)
|- (ph -> (ps -> ch))   =>   |- ((ph /\ ps) -> ch)
Theorem	impcom 349	Importation inference with commuted antecedents.
|- (ph -> (ps -> ch))   =>   |- ((ps /\ ph) -> ch)
Theorem	pm4.14 350	Theorem *4.14 of [WhiteheadRussell] p. 117.
|- (((ph /\ ps) -> ch) <-> ((ph /\ -. ch) -> -. ps))
Theorem	pm4.15 351	Theorem *4.15 of [WhiteheadRussell] p. 117.
|- (((ph /\ ps) -> -. ch) <-> ((ps /\ ch) -> -. ph))
Theorem	pm4.78 352	Theorem *4.78 of [WhiteheadRussell] p. 121.
|- (((ph -> ps) \/ (ph -> ch)) <-> (ph -> (ps \/ ch)))
Theorem	pm4.79 353	Theorem *4.79 of [WhiteheadRussell] p. 121.
|- (((ps -> ph) \/ (ch -> ph)) <-> ((ps /\ ch) -> ph))
Theorem	pm4.87 354	Theorem *4.87 of [WhiteheadRussell] p. 122. (The proof was shortened by Eric Schmidt, 26-Oct-2006.)
|- (((((ph /\ ps) -> ch) <-> (ph -> (ps -> ch))) /\ ((ph -> (ps -> ch)) <-> (ps -> (ph -> ch)))) /\ ((ps -> (ph -> ch)) <-> ((ps /\ ph) -> ch)))
Theorem	pm3.33 355	Theorem *3.33 (Syll) of [WhiteheadRussell] p. 112.
|- (((ph -> ps) /\ (ps -> ch)) -> (ph -> ch))
Theorem	pm3.34 356	Theorem *3.34 (Syll) of [WhiteheadRussell] p. 112.
|- (((ps -> ch) /\ (ph -> ps)) -> (ph -> ch))
Theorem	pm3.35 357	Conjunctive detachment. Theorem *3.35 of [WhiteheadRussell] p. 112.
|- ((ph /\ (ph -> ps)) -> ps)
Theorem	pm5.31 358	Theorem *5.31 of [WhiteheadRussell] p. 125.
|- ((ch /\ (ph -> ps)) -> (ph -> (ps /\ ch)))
Theorem	imp3a 359	Importation deduction.
|- (ph -> (ps -> (ch -> th)))   =>   |- (ph -> ((ps /\ ch) -> th))
Theorem	imp31 360	An importation inference.
|- (ph -> (ps -> (ch -> th)))   =>   |- (((ph /\ ps) /\ ch) -> th)
Theorem	imp32 361	An importation inference.
|- (ph -> (ps -> (ch -> th)))   =>   |- ((ph /\ (ps /\ ch)) -> th)
Theorem	imp4a 362	An importation inference.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (ph -> (ps -> ((ch /\ th) -> ta)))
Theorem	imp4b 363	An importation inference.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- ((ph /\ ps) -> ((ch /\ th) -> ta))
Theorem	imp4c 364	An importation inference.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (ph -> (((ps /\ ch) /\ th) -> ta))
Theorem	imp4d 365	An importation inference.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (ph -> ((ps /\ (ch /\ th)) -> ta))
Theorem	imp41 366	An importation inference.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- ((((ph /\ ps) /\ ch) /\ th) -> ta)
Theorem	imp42 367	An importation inference.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (((ph /\ (ps /\ ch)) /\ th) -> ta)
Theorem	imp43 368	An importation inference.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- (((ph /\ ps) /\ (ch /\ th)) -> ta)
Theorem	imp44 369	An importation inference.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- ((ph /\ ((ps /\ ch) /\ th)) -> ta)
Theorem	imp45 370	An importation inference.
|- (ph -> (ps -> (ch -> (th -> ta))))   =>   |- ((ph /\ (ps /\ (ch /\ th))) -> ta)
Theorem	ex 371	Exportation inference. (This theorem used to be labeled "exp" but was changed to "ex" so as not to conflict with the math token "exp", per the June 2006 Metamath spec change.) (The proof was shortened by Eric Schmidt, 22-Dec-2006.)
|- ((ph /\ ps) -> ch)   =>   |- (ph -> (ps -> ch))
Theorem	expcom 372	Exportation inference with commuted antecedents.
|- ((ph /\ ps) -> ch)   =>   |- (ps -> (ph -> ch))
Theorem	expimpd 373	Exportation followed by a deduction version of importation.
|- ((ph /\ ps) -> (ch -> th))   =>   |- (ph -> ((ps /\ ch) -> th))
Theorem	exp3a 374	Exportation deduction.
|- (ph -> ((ps /\ ch) -> th))   =>   |- (ph -> (ps -> (ch -> th)))
Theorem	expdimp 375	A deduction version of exportation, followed by importation.
|- (ph -> ((ps /\ ch) -> th))   =>   |- ((ph /\ ps) -> (ch -> th))
Theorem	exp31 376	An exportation inference.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (ph -> (ps -> (ch -> th)))
Theorem	exp32 377	An exportation inference.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- (ph -> (ps -> (ch -> th)))
Theorem	exp4a 378	An exportation inference.
|- (ph -> (ps -> ((ch /\ th) -> ta)))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp4b 379	An exportation inference.
|- ((ph /\ ps) -> ((ch /\ th) -> ta))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp4c 380	An exportation inference.
|- (ph -> (((ps /\ ch) /\ th) -> ta))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp4d 381	An exportation inference.
|- (ph -> ((ps /\ (ch /\ th)) -> ta))   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp41 382	An exportation inference.
|- ((((ph /\ ps) /\ ch) /\ th) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp42 383	An exportation inference.
|- (((ph /\ (ps /\ ch)) /\ th) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp43 384	An exportation inference.
|- (((ph /\ ps) /\ (ch /\ th)) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp44 385	An exportation inference.
|- ((ph /\ ((ps /\ ch) /\ th)) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	exp45 386	An exportation inference.
|- ((ph /\ (ps /\ (ch /\ th))) -> ta)   =>   |- (ph -> (ps -> (ch -> (th -> ta))))
Theorem	impac 387	Importation with conjunction in consequent.
|- (ph -> (ps -> ch))   =>   |- ((ph /\ ps) -> (ch /\ ps))
Theorem	adantl 388	Inference adding a conjunct to the left of an antecedent.
|- (ph -> ps)   =>   |- ((ch /\ ph) -> ps)
Theorem	adantr 389	Inference adding a conjunct to the right of an antecedent.
|- (ph -> ps)   =>   |- ((ph /\ ch) -> ps)
Theorem	adantld 390	Deduction adding a conjunct to the left of an antecedent.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((th /\ ps) -> ch))
Theorem	adantrd 391	Deduction adding a conjunct to the right of an antecedent.
|- (ph -> (ps -> ch))   =>   |- (ph -> ((ps /\ th) -> ch))
Theorem	adantll 392	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((th /\ ph) /\ ps) -> ch)
Theorem	adantlr 393	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- (((ph /\ th) /\ ps) -> ch)
Theorem	adantrl 394	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- ((ph /\ (th /\ ps)) -> ch)
Theorem	adantrr 395	Deduction adding a conjunct to antecedent.
|- ((ph /\ ps) -> ch)   =>   |- ((ph /\ (ps /\ th)) -> ch)
Theorem	adantlll 396	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((((ta /\ ph) /\ ps) /\ ch) -> th)
Theorem	adantllr 397	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- ((((ph /\ ta) /\ ps) /\ ch) -> th)
Theorem	adantlrl 398	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ (ta /\ ps)) /\ ch) -> th)
Theorem	adantlrr 399	Deduction adding a conjunct to antecedent.
|- (((ph /\ ps) /\ ch) -> th)   =>   |- (((ph /\ (ps /\ ta)) /\ ch) -> th)
Theorem	adantrll 400	Deduction adding a conjunct to antecedent.
|- ((ph /\ (ps /\ ch)) -> th)   =>   |- ((ph /\ ((ta /\ ps) /\ ch)) -> th)