mmtheorems3 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 201-300 - Page 3 of 123

Type	Label	Description
Statement
Theorem	sylbid 201	A syllogism deduction.
|- (ph -> (ps <-> ch))   &   |- (ph -> (ch -> th))   =>   |- (ph -> (ps -> th))
Theorem	sylibrd 202	A syllogism deduction.
|- (ph -> (ps -> ch))   &   |- (ph -> (th <-> ch))   =>   |- (ph -> (ps -> th))
Theorem	sylbird 203	A syllogism deduction.
|- (ph -> (ch <-> ps))   &   |- (ph -> (ch -> th))   =>   |- (ph -> (ps -> th))
Theorem	syl5ib 204	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition.
|- (ph -> (ps -> ch))   &   |- (th <-> ps)   =>   |- (ph -> (th -> ch))
Theorem	syl5ibr 205	A mixed syllogism inference from a nested implication and a biconditional.
|- (ph -> (ps -> ch))   &   |- (ps <-> th)   =>   |- (ph -> (th -> ch))
Theorem	syl5bi 206	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (th -> ps)   =>   |- (ph -> (th -> ch))
Theorem	syl5cbi 207	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (th -> ps)   =>   |- (th -> (ph -> ch))
Theorem	syl5bir 208	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (th -> ch)   =>   |- (ph -> (th -> ps))
Theorem	syl5cbir 209	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (th -> ch)   =>   |- (th -> (ph -> ps))
Theorem	syl6ib 210	A mixed syllogism inference from a nested implication and a biconditional.
|- (ph -> (ps -> ch))   &   |- (ch <-> th)   =>   |- (ph -> (ps -> th))
Theorem	syl6ibr 211	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition.
|- (ph -> (ps -> ch))   &   |- (th <-> ch)   =>   |- (ph -> (ps -> th))
Theorem	syl6bi 212	A mixed syllogism inference.
|- (ph -> (ps <-> ch))   &   |- (ch -> th)   =>   |- (ph -> (ps -> th))
Theorem	syl6bir 213	A mixed syllogism inference.
|- (ph -> (ch <-> ps))   &   |- (ch -> th)   =>   |- (ph -> (ps -> th))
Theorem	syl7ib 214	A mixed syllogism inference from a doubly nested implication and a biconditional.
|- (ph -> (ps -> (ch -> th)))   &   |- (ta <-> ch)   =>   |- (ph -> (ps -> (ta -> th)))
Theorem	syl8ib 215	A syllogism rule of inference. The second premise is used to replace the consequent of the first premise.
|- (ph -> (ps -> (ch -> th)))   &   |- (th <-> ta)   =>   |- (ph -> (ps -> (ch -> ta)))
Theorem	3imtr3i 216	A mixed syllogism inference, useful for removing a definition from both sides of an implication.
|- (ph -> ps)   &   |- (ph <-> ch)   &   |- (ps <-> th)   =>   |- (ch -> th)
Theorem	3imtr4i 217	A mixed syllogism inference, useful for applying a definition to both sides of an implication.
|- (ph -> ps)   &   |- (ch <-> ph)   &   |- (th <-> ps)   =>   |- (ch -> th)
Theorem	con1bii 218	A contraposition inference.
|- (-. ph <-> ps)   =>   |- (-. ps <-> ph)
Theorem	con2bii 219	A contraposition inference.
|- (ph <-> -. ps)   =>   |- (ps <-> -. ph)
Logical disjunction and conjunction
Syntax	wo 220	Extend wff definition to include disjunction ('or').
wff (ph \/ ps)
Syntax	wa 221	Extend wff definition to include conjunction ('and').
wff (ph /\ ps)
Definition	df-or 222	Define disjunction (logical 'or'). This is our first use of the biconditional connective in a definition; we use it in place of the traditional "<=def=>", which means the same thing, except that we can manipulate the biconditional connective directly in proofs rather than having to rely on an informal definition substitution rule. Note that if we mechanically substitute (-. ph -> ps) for (ph \/ ps), we end up with an instance of previously proved theorem biid 168. This is the justification for the definition, along with the fact that it introduces a new symbol \/. Definition of [Margaris] p. 49.
|- ((ph \/ ps) <-> (-. ph -> ps))
Definition	df-an 223	Define conjunction (logical 'and'). Definition of [Margaris] p. 49.
|- ((ph /\ ps) <-> -. (ph -> -. ps))
Theorem	pm4.64 224	Theorem *4.64 of [WhiteheadRussell] p. 120.
|- ((-. ph -> ps) <-> (ph \/ ps))
Theorem	pm2.54 225	Theorem *2.54 of [WhiteheadRussell] p. 107.
|- ((-. ph -> ps) -> (ph \/ ps))
Theorem	pm4.63 226	Theorem *4.63 of [WhiteheadRussell] p. 120.
|- (-. (ph -> -. ps) <-> (ph /\ ps))
Theorem	dfor2 227	Logical 'or' expressed in terms of implication only. Theorem *5.25 of [WhiteheadRussell] p. 124.
|- ((ph \/ ps) <-> ((ph -> ps) -> ps))
Theorem	ori 228	Infer implication from disjunction.
|- (ph \/ ps)   =>   |- (-. ph -> ps)
Theorem	orri 229	Infer implication from disjunction.
|- (-. ph -> ps)   =>   |- (ph \/ ps)
Theorem	ord 230	Deduce implication from disjunction.
|- (ph -> (ps \/ ch))   =>   |- (ph -> (-. ps -> ch))
Theorem	orrd 231	Deduce implication from disjunction.
|- (ph -> (-. ps -> ch))   =>   |- (ph -> (ps \/ ch))
Theorem	imor 232	Implication in terms of disjunction. Theorem *4.6 of [WhiteheadRussell] p. 120.
|- ((ph -> ps) <-> (-. ph \/ ps))
Theorem	pm4.62 233	Theorem *4.62 of [WhiteheadRussell] p. 120.
|- ((ph -> -. ps) <-> (-. ph \/ -. ps))
Theorem	pm4.66 234	Theorem *4.66 of [WhiteheadRussell] p. 120.
|- ((-. ph -> -. ps) <-> (ph \/ -. ps))
Theorem	iman 235	Express implication in terms of conjunction. Theorem 3.4(27) of [Stoll] p. 176.
|- ((ph -> ps) <-> -. (ph /\ -. ps))
Theorem	annim 236	Express conjunction in terms of implication.
|- ((ph /\ -. ps) <-> -. (ph -> ps))
Theorem	pm4.61 237	Theorem *4.61 of [WhiteheadRussell] p. 120.
|- (-. (ph -> ps) <-> (ph /\ -. ps))
Theorem	pm4.65 238	Theorem *4.65 of [WhiteheadRussell] p. 120.
|- (-. (-. ph -> ps) <-> (-. ph /\ -. ps))
Theorem	pm4.67 239	Theorem *4.67 of [WhiteheadRussell] p. 120.
|- (-. (-. ph -> -. ps) <-> (-. ph /\ ps))
Theorem	imnan 240	Express implication in terms of conjunction.
|- ((ph -> -. ps) <-> -. (ph /\ ps))
Theorem	oridm 241	Idempotent law for disjunction. Theorem *4.25 of [WhiteheadRussell] p. 117.
|- ((ph \/ ph) <-> ph)
Theorem	pm4.25 242	Theorem *4.25 of [WhiteheadRussell] p. 117.
|- (ph <-> (ph \/ ph))
Theorem	pm1.2 243	Axiom *1.2 (Taut) of [WhiteheadRussell] p. 96.
|- ((ph \/ ph) -> ph)
Theorem	orcom 244	Commutative law for disjunction. Theorem *4.31 of [WhiteheadRussell] p. 118.
|- ((ph \/ ps) <-> (ps \/ ph))
Theorem	pm1.4 245	Axiom *1.4 of [WhiteheadRussell] p. 96.
|- ((ph \/ ps) -> (ps \/ ph))
Theorem	pm2.62 246	Theorem *2.62 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> ((ph -> ps) -> ps))
Theorem	pm2.621 247	Theorem *2.621 of [WhiteheadRussell] p. 107.
|- ((ph -> ps) -> ((ph \/ ps) -> ps))
Theorem	pm2.68 248	Theorem *2.68 of [WhiteheadRussell] p. 108.
|- (((ph -> ps) -> ps) -> (ph \/ ps))
Theorem	orel1 249	Elimination of disjunction by denial of a disjunct. Theorem *2.55 of [WhiteheadRussell] p. 107.
|- (-. ph -> ((ph \/ ps) -> ps))
Theorem	orel2 250	Elimination of disjunction by denial of a disjunct. Theorem *2.56 of [WhiteheadRussell] p. 107.
|- (-. ph -> ((ps \/ ph) -> ps))
Theorem	pm2.25 251	Theorem *2.25 of [WhiteheadRussell] p. 104.
|- (ph \/ ((ph \/ ps) -> ps))
Theorem	pm2.53 252	Theorem *2.53 of [WhiteheadRussell] p. 107.
|- ((ph \/ ps) -> (-. ph -> ps))
Theorem	orbi2i 253	Inference adding a left disjunct to both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ((ch \/ ph) <-> (ch \/ ps))
Theorem	orbi1i 254	Inference adding a right disjunct to both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ((ph \/ ch) <-> (ps \/ ch))
Theorem	orbi12i 255	Infer the disjunction of two equivalences.
|- (ph <-> ps)   &   |- (ch <-> th)   =>   |- ((ph \/ ch) <-> (ps \/ th))
Theorem	or12 256	A rearrangement of disjuncts.
|- ((ph \/ (ps \/ ch)) <-> (ps \/ (ph \/ ch)))
Theorem	pm1.5 257	Axiom *1.5 (Assoc) of [WhiteheadRussell] p. 96.
|- ((ph \/ (ps \/ ch)) -> (ps \/ (ph \/ ch)))
Theorem	orass 258	Associative law for disjunction. Theorem *4.33 of [WhiteheadRussell] p. 118.
|- (((ph \/ ps) \/ ch) <-> (ph \/ (ps \/ ch)))
Theorem	pm2.31 259	Theorem *2.31 of [WhiteheadRussell] p. 104.
|- ((ph \/ (ps \/ ch)) -> ((ph \/ ps) \/ ch))
Theorem	pm2.32 260	Theorem *2.32 of [WhiteheadRussell] p. 105.
|- (((ph \/ ps) \/ ch) -> (ph \/ (ps \/ ch)))
Theorem	or23 261	A rearrangement of disjuncts.
|- (((ph \/ ps) \/ ch) <-> ((ph \/ ch) \/ ps))
Theorem	or4 262	Rearrangement of 4 disjuncts.
|- (((ph \/ ps) \/ (ch \/ th)) <-> ((ph \/ ch) \/ (ps \/ th)))
Theorem	or42 263	Rearrangement of 4 disjuncts.
|- (((ph \/ ps) \/ (ch \/ th)) <-> ((ph \/ ch) \/ (th \/ ps)))
Theorem	orordi 264	Distribution of disjunction over disjunction.
|- ((ph \/ (ps \/ ch)) <-> ((ph \/ ps) \/ (ph \/ ch)))
Theorem	orordir 265	Distribution of disjunction over disjunction.
|- (((ph \/ ps) \/ ch) <-> ((ph \/ ch) \/ (ps \/ ch)))
Theorem	olc 266	Introduction of a disjunct. Axiom *1.3 of [WhiteheadRussell] p. 96.
|- (ph -> (ps \/ ph))
Theorem	orc 267	Introduction of a disjunct. Theorem *2.2 of [WhiteheadRussell] p. 104.
|- (ph -> (ph \/ ps))
Theorem	orci 268	Deduction introducing a disjunct.
|- ph   =>   |- (ph \/ ps)
Theorem	olci 269	Deduction introducing a disjunct.
|- ph   =>   |- (ps \/ ph)
Theorem	orcd 270	Deduction introducing a disjunct.
|- (ph -> ps)   =>   |- (ph -> (ps \/ ch))
Theorem	olcd 271	Deduction introducing a disjunct.
|- (ph -> ps)   =>   |- (ph -> (ch \/ ps))
Theorem	orcs 272	Deduction eliminating disjunct.
|- ((ph \/ ps) -> ch)   =>   |- (ph -> ch)
Theorem	olcs 273	Deduction eliminating disjunct.
|- ((ph \/ ps) -> ch)   =>   |- (ps -> ch)
Theorem	pm2.07 274	Theorem *2.07 of [WhiteheadRussell] p. 101.
|- (ph -> (ph \/ ph))
Theorem	pm2.45 275	Theorem *2.45 of [WhiteheadRussell] p. 106.
|- (-. (ph \/ ps) -> -. ph)
Theorem	pm2.46 276	Theorem *2.46 of [WhiteheadRussell] p. 106.
|- (-. (ph \/ ps) -> -. ps)
Theorem	pm2.47 277	Theorem *2.47 of [WhiteheadRussell] p. 107.
|- (-. (ph \/ ps) -> (-. ph \/ ps))
Theorem	pm2.48 278	Theorem *2.48 of [WhiteheadRussell] p. 107.
|- (-. (ph \/ ps) -> (ph \/ -. ps))
Theorem	pm2.49 279	Theorem *2.49 of [WhiteheadRussell] p. 107.
|- (-. (ph \/ ps) -> (-. ph \/ -. ps))
Theorem	pm2.67 280	Theorem *2.67 of [WhiteheadRussell] p. 107.
|- (((ph \/ ps) -> ps) -> (ph -> ps))
Theorem	pm3.2 281	Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell] p. 111.
|- (ph -> (ps -> (ph /\ ps)))
Theorem	pm3.21 282	Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111.
|- (ph -> (ps -> (ps /\ ph)))
Theorem	pm3.2i 283	Infer conjunction of premises.
|- ph   &   |- ps   =>   |- (ph /\ ps)
Theorem	pm3.37 284	Theorem *3.37 (Transp) of [WhiteheadRussell] p. 112.
|- (((ph /\ ps) -> ch) -> ((ph /\ -. ch) -> -. ps))
Theorem	pm3.43i 285	Nested conjunction of antecedents.
|- ((ph -> ps) -> ((ph -> ch) -> (ph -> (ps /\ ch))))
Theorem	jca 286	Deduce conjunction of the consequents of two implications ("join consequents with 'and'").
|- (ph -> ps)   &   |- (ph -> ch)   =>   |- (ph -> (ps /\ ch))
Theorem	jcai 287	Deduction replacing implication with conjunction.
|- (ph -> ps)   &   |- (ph -> (ps -> ch))   =>   |- (ph -> (ps /\ ch))
Theorem	jctl 288	Inference conjoining a theorem to the left of a consequent.
|- ps   =>   |- (ph -> (ps /\ ph))
Theorem	jctr 289	Inference conjoining a theorem to the right of a consequent.
|- ps   =>   |- (ph -> (ph /\ ps))
Theorem	jctil 290	Inference conjoining a theorem to left of consequent in an implication.
|- (ph -> ps)   &   |- ch   =>   |- (ph -> (ch /\ ps))
Theorem	jctir 291	Inference conjoining a theorem to right of consequent in an implication.
|- (ph -> ps)   &   |- ch   =>   |- (ph -> (ps /\ ch))
Theorem	ancl 292	Conjoin antecedent to left of consequent.
|- ((ph -> ps) -> (ph -> (ph /\ ps)))
Theorem	ancr 293	Conjoin antecedent to right of consequent.
|- ((ph -> ps) -> (ph -> (ps /\ ph)))
Theorem	ancli 294	Deduction conjoining antecedent to left of consequent.
|- (ph -> ps)   =>   |- (ph -> (ph /\ ps))
Theorem	ancri 295	Deduction conjoining antecedent to right of consequent.
|- (ph -> ps)   =>   |- (ph -> (ps /\ ph))
Theorem	ancld 296	Deduction conjoining antecedent to left of consequent in nested implication.
|- (ph -> (ps -> ch))   =>   |- (ph -> (ps -> (ps /\ ch)))
Theorem	ancrd 297	Deduction conjoining antecedent to right of consequent in nested implication.
|- (ph -> (ps -> ch))   =>   |- (ph -> (ps -> (ch /\ ps)))
Theorem	anc2l 298	Conjoin antecedent to left of consequent in nested implication.
|- ((ph -> (ps -> ch)) -> (ph -> (ps -> (ph /\ ch))))
Theorem	anc2r 299	Conjoin antecedent to right of consequent in nested implication.
|- ((ph -> (ps -> ch)) -> (ph -> (ps -> (ch /\ ph))))
Theorem	anc2li 300	Deduction conjoining antecedent to left of consequent in nested implication.
|- (ph -> (ps -> ch))   =>   |- (ph -> (ps -> (ph /\ ch)))