P. 3 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 201-300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	3imtr3d 201	More general version of 3imtr3i 199. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( ps <-> th ) )   &    |- ( ph -> ( ch <-> ta ) )   =>    |- ( ph -> ( th -> ta ) )
Theorem	3imtr4d 202	More general version of 3imtr4i 200. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th <-> ps ) )   &    |- ( ph -> ( ta <-> ch ) )   =>    |- ( ph -> ( th -> ta ) )
Theorem	3imtr3g 203	More general version of 3imtr3i 199. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ps <-> th )   &    |- ( ch <-> ta )   =>    |- ( ph -> ( th -> ta ) )
Theorem	3imtr4g 204	More general version of 3imtr4i 200. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th <-> ps )   &    |- ( ta <-> ch )   =>    |- ( ph -> ( th -> ta ) )
Theorem	3bitri 205	A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ps <-> ch )   &    |- ( ch <-> th )   =>    |- ( ph <-> th )
Theorem	3bitrri 206	A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
|- ( ph <-> ps )   &    |- ( ps <-> ch )   &    |- ( ch <-> th )   =>    |- ( th <-> ph )
Theorem	3bitr2i 207	A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
|- ( ph <-> ps )   &    |- ( ch <-> ps )   &    |- ( ch <-> th )   =>    |- ( ph <-> th )
Theorem	3bitr2ri 208	A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
|- ( ph <-> ps )   &    |- ( ch <-> ps )   &    |- ( ch <-> th )   =>    |- ( th <-> ph )
Theorem	3bitr3i 209	A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ph <-> ch )   &    |- ( ps <-> th )   =>    |- ( ch <-> th )
Theorem	3bitr3ri 210	A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ph <-> ch )   &    |- ( ps <-> th )   =>    |- ( th <-> ch )
Theorem	3bitr4i 211	A chained inference from transitive law for logical equivalence. This inference is frequently used to apply a definition to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch <-> ph )   &    |- ( th <-> ps )   =>    |- ( ch <-> th )
Theorem	3bitr4ri 212	A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
|- ( ph <-> ps )   &    |- ( ch <-> ph )   &    |- ( th <-> ps )   =>    |- ( th <-> ch )
Theorem	3bitrd 213	Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( ch <-> th ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ps <-> ta ) )
Theorem	3bitrrd 214	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( ch <-> th ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ta <-> ps ) )
Theorem	3bitr2d 215	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ps <-> ta ) )
Theorem	3bitr2rd 216	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ta <-> ps ) )
Theorem	3bitr3d 217	Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 24-Apr-1996.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( ps <-> th ) )   &    |- ( ph -> ( ch <-> ta ) )   =>    |- ( ph -> ( th <-> ta ) )
Theorem	3bitr3rd 218	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( ps <-> th ) )   &    |- ( ph -> ( ch <-> ta ) )   =>    |- ( ph -> ( ta <-> th ) )
Theorem	3bitr4d 219	Deduction from transitivity of biconditional. Useful for converting conditional definitions in a formula. (Contributed by NM, 18-Oct-1995.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ps ) )   &    |- ( ph -> ( ta <-> ch ) )   =>    |- ( ph -> ( th <-> ta ) )
Theorem	3bitr4rd 220	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ps ) )   &    |- ( ph -> ( ta <-> ch ) )   =>    |- ( ph -> ( ta <-> th ) )
Theorem	3bitr3g 221	More general version of 3bitr3i 209. Useful for converting definitions in a formula. (Contributed by NM, 4-Jun-1995.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ps <-> th )   &    |- ( ch <-> ta )   =>    |- ( ph -> ( th <-> ta ) )
Theorem	3bitr4g 222	More general version of 3bitr4i 211. Useful for converting definitions in a formula. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( th <-> ps )   &    |- ( ta <-> ch )   =>    |- ( ph -> ( th <-> ta ) )
Theorem	bi3ant 223	Construct a biconditional in antecedent position. (Contributed by Wolf Lammen, 14-May-2013.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ( ( th -> ta ) -> ph ) -> ( ( ( ta -> th ) -> ps ) -> ( ( th <-> ta ) -> ch ) ) )
Theorem	bisym 224	Express symmetries of theorems in terms of biconditionals. (Contributed by Wolf Lammen, 14-May-2013.)
|- ( ( ( ph -> ps ) -> ( ch -> th ) ) -> ( ( ( ps -> ph ) -> ( th -> ch ) ) -> ( ( ph <-> ps ) -> ( ch <-> th ) ) ) )
Theorem	imbi2i 225	Introduce an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 6-Feb-2013.)
|- ( ph <-> ps )   =>    |- ( ( ch -> ph ) <-> ( ch -> ps ) )
Theorem	bibi2i 226	Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.)
|- ( ph <-> ps )   =>    |- ( ( ch <-> ph ) <-> ( ch <-> ps ) )
Theorem	bibi1i 227	Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- ( ( ph <-> ch ) <-> ( ps <-> ch ) )
Theorem	bibi12i 228	The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph <-> ch ) <-> ( ps <-> th ) )
Theorem	imbi2d 229	Deduction adding an antecedent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th -> ps ) <-> ( th -> ch ) ) )
Theorem	imbi1d 230	Deduction adding a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps -> th ) <-> ( ch -> th ) ) )
Theorem	bibi2d 231	Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( th <-> ps ) <-> ( th <-> ch ) ) )
Theorem	bibi1d 232	Deduction adding a biconditional to the right in an equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> th ) ) )
Theorem	imbi12d 233	Deduction joining two equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps -> th ) <-> ( ch -> ta ) ) )
Theorem	bibi12d 234	Deduction joining two equivalences to form equivalence of biconditionals. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ta ) )   =>    |- ( ph -> ( ( ps <-> th ) <-> ( ch <-> ta ) ) )
Theorem	imbi1 235	Theorem *4.84 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph <-> ps ) -> ( ( ph -> ch ) <-> ( ps -> ch ) ) )
Theorem	imbi2 236	Theorem *4.85 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 19-May-2013.)
|- ( ( ph <-> ps ) -> ( ( ch -> ph ) <-> ( ch -> ps ) ) )
Theorem	imbi1i 237	Introduce a consequent to both sides of a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 17-Sep-2013.)
|- ( ph <-> ps )   =>    |- ( ( ph -> ch ) <-> ( ps -> ch ) )
Theorem	imbi12i 238	Join two logical equivalences to form equivalence of implications. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch <-> th )   =>    |- ( ( ph -> ch ) <-> ( ps -> th ) )
Theorem	bibi1 239	Theorem *4.86 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.)
|- ( ( ph <-> ps ) -> ( ( ph <-> ch ) <-> ( ps <-> ch ) ) )
Theorem	biimt 240	A wff is equivalent to itself with true antecedent. (Contributed by NM, 28-Jan-1996.)
|- ( ph -> ( ps <-> ( ph -> ps ) ) )
Theorem	pm5.5 241	Theorem *5.5 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
|- ( ph -> ( ( ph -> ps ) <-> ps ) )
Theorem	a1bi 242	Inference introducing a theorem as an antecedent. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
|- ph   =>    |- ( ps <-> ( ph -> ps ) )
Theorem	pm5.501 243	Theorem *5.501 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 24-Jan-2013.)
|- ( ph -> ( ps <-> ( ph <-> ps ) ) )
Theorem	ibib 244	Implication in terms of implication and biconditional. (Contributed by NM, 31-Mar-1994.) (Proof shortened by Wolf Lammen, 24-Jan-2013.)
|- ( ( ph -> ps ) <-> ( ph -> ( ph <-> ps ) ) )
Theorem	ibibr 245	Implication in terms of implication and biconditional. (Contributed by NM, 29-Apr-2005.) (Proof shortened by Wolf Lammen, 21-Dec-2013.)
|- ( ( ph -> ps ) <-> ( ph -> ( ps <-> ph ) ) )
Theorem	tbt 246	A wff is equivalent to its equivalence with truth. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
|- ph   =>    |- ( ps <-> ( ps <-> ph ) )
Theorem	bi2.04 247	Logical equivalence of commuted antecedents. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.)
|- ( ( ph -> ( ps -> ch ) ) <-> ( ps -> ( ph -> ch ) ) )
Theorem	pm5.4 248	Antecedent absorption implication. Theorem *5.4 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
|- ( ( ph -> ( ph -> ps ) ) <-> ( ph -> ps ) )
Theorem	imdi 249	Distributive law for implication. Compare Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 5-Aug-1993.)
|- ( ( ph -> ( ps -> ch ) ) <-> ( ( ph -> ps ) -> ( ph -> ch ) ) )
Theorem	pm5.41 250	Theorem *5.41 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 12-Oct-2012.)
|- ( ( ( ph -> ps ) -> ( ph -> ch ) ) <-> ( ph -> ( ps -> ch ) ) )
Theorem	imim21b 251	Simplify an implication between two implications when the antecedent of the first is a consequence of the antecedent of the second. The reverse form is useful in producing the successor step in induction proofs. (Contributed by Paul Chapman, 22-Jun-2011.) (Proof shortened by Wolf Lammen, 14-Sep-2013.)
|- ( ( ps -> ph ) -> ( ( ( ph -> ch ) -> ( ps -> th ) ) <-> ( ps -> ( ch -> th ) ) ) )
Theorem	impd 252	Importation deduction. (Contributed by NM, 31-Mar-1994.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   =>    |- ( ph -> ( ( ps /\ ch ) -> th ) )
Theorem	impcomd 253	Importation deduction with commuted antecedents. (Contributed by Peter Mazsa, 24-Sep-2022.) (Proof shortened by Wolf Lammen, 22-Oct-2022.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   =>    |- ( ph -> ( ( ch /\ ps ) -> th ) )
Theorem	imp31 254	An importation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   =>    |- ( ( ( ph /\ ps ) /\ ch ) -> th )
Theorem	imp32 255	An importation inference. (Contributed by NM, 26-Apr-1994.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   =>    |- ( ( ph /\ ( ps /\ ch ) ) -> th )
Theorem	expd 256	Exportation deduction. (Contributed by NM, 20-Aug-1993.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ps -> ( ch -> th ) ) )
Theorem	expdimp 257	A deduction version of exportation, followed by importation. (Contributed by NM, 6-Sep-2008.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ( ph /\ ps ) -> ( ch -> th ) )
Theorem	impancom 258	Mixed importation/commutation inference. (Contributed by NM, 22-Jun-2013.)
|- ( ( ph /\ ps ) -> ( ch -> th ) )   =>    |- ( ( ph /\ ch ) -> ( ps -> th ) )
Theorem	pm3.3 259	Theorem *3.3 (Exp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
|- ( ( ( ph /\ ps ) -> ch ) -> ( ph -> ( ps -> ch ) ) )
Theorem	pm3.31 260	Theorem *3.31 (Imp) of [WhiteheadRussell] p. 112. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
|- ( ( ph -> ( ps -> ch ) ) -> ( ( ph /\ ps ) -> ch ) )
Theorem	impexp 261	Import-export theorem. Part of Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 24-Mar-2013.)
|- ( ( ( ph /\ ps ) -> ch ) <-> ( ph -> ( ps -> ch ) ) )
Theorem	pm3.21 262	Join antecedents with conjunction. Theorem *3.21 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps -> ( ps /\ ph ) ) )
Theorem	pm3.22 263	Theorem *3.22 of [WhiteheadRussell] p. 111. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
|- ( ( ph /\ ps ) -> ( ps /\ ph ) )
Theorem	ancom 264	Commutative law for conjunction. Theorem *4.3 of [WhiteheadRussell] p. 118. (Contributed by NM, 25-Jun-1998.) (Proof shortened by Wolf Lammen, 4-Nov-2012.)
|- ( ( ph /\ ps ) <-> ( ps /\ ph ) )
Theorem	ancomd 265	Commutation of conjuncts in consequent. (Contributed by Jeff Hankins, 14-Aug-2009.)
|- ( ph -> ( ps /\ ch ) )   =>    |- ( ph -> ( ch /\ ps ) )
Theorem	ancoms 266	Inference commuting conjunction in antecedent. (Contributed by NM, 21-Apr-1994.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ( ps /\ ph ) -> ch )
Theorem	ancomsd 267	Deduction commuting conjunction in antecedent. (Contributed by NM, 12-Dec-2004.)
|- ( ph -> ( ( ps /\ ch ) -> th ) )   =>    |- ( ph -> ( ( ch /\ ps ) -> th ) )
Theorem	biancomi 268	Commuting conjunction in a biconditional. (Contributed by Peter Mazsa, 17-Jun-2018.)
|- ( ph <-> ( ch /\ ps ) )   =>    |- ( ph <-> ( ps /\ ch ) )
Theorem	biancomd 269	Commuting conjunction in a biconditional, deduction form. (Contributed by Peter Mazsa, 3-Oct-2018.)
|- ( ph -> ( ps <-> ( th /\ ch ) ) )   =>    |- ( ph -> ( ps <-> ( ch /\ th ) ) )
Theorem	pm3.2i 270	Infer conjunction of premises. (Contributed by NM, 5-Aug-1993.)
|- ph   &    |- ps   =>    |- ( ph /\ ps )
Theorem	pm3.43i 271	Nested conjunction of antecedents. (Contributed by NM, 5-Aug-1993.)
|- ( ( ph -> ps ) -> ( ( ph -> ch ) -> ( ph -> ( ps /\ ch ) ) ) )
Theorem	simplbi 272	Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ph -> ps )
Theorem	simprbi 273	Deduction eliminating a conjunct. (Contributed by NM, 27-May-1998.)
|- ( ph <-> ( ps /\ ch ) )   =>    |- ( ph -> ch )
Theorem	adantr 274	Inference adding a conjunct to the right of an antecedent. (Contributed by NM, 30-Aug-1993.)
|- ( ph -> ps )   =>    |- ( ( ph /\ ch ) -> ps )
Theorem	adantl 275	Inference adding a conjunct to the left of an antecedent. (Contributed by NM, 30-Aug-1993.) (Proof shortened by Wolf Lammen, 23-Nov-2012.)
|- ( ph -> ps )   =>    |- ( ( ch /\ ph ) -> ps )
Theorem	adantld 276	Deduction adding a conjunct to the left of an antecedent. (Contributed by NM, 4-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2012.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( th /\ ps ) -> ch ) )
Theorem	adantrd 277	Deduction adding a conjunct to the right of an antecedent. (Contributed by NM, 4-May-1994.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ph -> ( ( ps /\ th ) -> ch ) )
Theorem	impel 278	An inference for implication elimination. (Contributed by Giovanni Mascellani, 23-May-2019.) (Proof shortened by Wolf Lammen, 2-Sep-2020.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th -> ps )   =>    |- ( ( ph /\ th ) -> ch )
Theorem	mpan9 279	Modus ponens conjoining dissimilar antecedents. (Contributed by NM, 1-Feb-2008.) (Proof shortened by Andrew Salmon, 7-May-2011.)
|- ( ph -> ps )   &    |- ( ch -> ( ps -> th ) )   =>    |- ( ( ph /\ ch ) -> th )
Theorem	syldan 280	A syllogism deduction with conjoined antecents. (Contributed by NM, 24-Feb-2005.) (Proof shortened by Wolf Lammen, 6-Apr-2013.)
|- ( ( ph /\ ps ) -> ch )   &    |- ( ( ph /\ ch ) -> th )   =>    |- ( ( ph /\ ps ) -> th )
Theorem	sylan 281	A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
|- ( ph -> ps )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ph /\ ch ) -> th )
Theorem	sylanb 282	A syllogism inference. (Contributed by NM, 18-May-1994.)
|- ( ph <-> ps )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ph /\ ch ) -> th )
Theorem	sylanbr 283	A syllogism inference. (Contributed by NM, 18-May-1994.)
|- ( ps <-> ph )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ph /\ ch ) -> th )
Theorem	sylan2 284	A syllogism inference. (Contributed by NM, 21-Apr-1994.) (Proof shortened by Wolf Lammen, 22-Nov-2012.)
|- ( ph -> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ps /\ ph ) -> th )
Theorem	sylan2b 285	A syllogism inference. (Contributed by NM, 21-Apr-1994.)
|- ( ph <-> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ps /\ ph ) -> th )
Theorem	sylan2br 286	A syllogism inference. (Contributed by NM, 21-Apr-1994.)
|- ( ch <-> ph )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ps /\ ph ) -> th )
Theorem	syl2an 287	A double syllogism inference. (Contributed by NM, 31-Jan-1997.)
|- ( ph -> ps )   &    |- ( ta -> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ph /\ ta ) -> th )
Theorem	syl2anr 288	A double syllogism inference. (Contributed by NM, 17-Sep-2013.)
|- ( ph -> ps )   &    |- ( ta -> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ta /\ ph ) -> th )
Theorem	syl2anb 289	A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
|- ( ph <-> ps )   &    |- ( ta <-> ch )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ph /\ ta ) -> th )
Theorem	syl2anbr 290	A double syllogism inference. (Contributed by NM, 29-Jul-1999.)
|- ( ps <-> ph )   &    |- ( ch <-> ta )   &    |- ( ( ps /\ ch ) -> th )   =>    |- ( ( ph /\ ta ) -> th )
Theorem	syland 291	A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( ( ch /\ th ) -> ta ) )   =>    |- ( ph -> ( ( ps /\ th ) -> ta ) )
Theorem	sylan2d 292	A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( ( th /\ ch ) -> ta ) )   =>    |- ( ph -> ( ( th /\ ps ) -> ta ) )
Theorem	syl2and 293	A syllogism deduction. (Contributed by NM, 15-Dec-2004.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th -> ta ) )   &    |- ( ph -> ( ( ch /\ ta ) -> et ) )   =>    |- ( ph -> ( ( ps /\ th ) -> et ) )
Theorem	biimpa 294	Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ( ph /\ ps ) -> ch )
Theorem	biimpar 295	Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ( ph /\ ch ) -> ps )
Theorem	biimpac 296	Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ( ps /\ ph ) -> ch )
Theorem	biimparc 297	Inference from a logical equivalence. (Contributed by NM, 3-May-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ( ch /\ ph ) -> ps )
Theorem	iba 298	Introduction of antecedent as conjunct. Theorem *4.73 of [WhiteheadRussell] p. 121. (Contributed by NM, 30-Mar-1994.) (Revised by NM, 24-Mar-2013.)
|- ( ph -> ( ps <-> ( ps /\ ph ) ) )
Theorem	ibar 299	Introduction of antecedent as conjunct. (Contributed by NM, 5-Dec-1995.) (Revised by NM, 24-Mar-2013.)
|- ( ph -> ( ps <-> ( ph /\ ps ) ) )
Theorem	biantru 300	A wff is equivalent to its conjunction with truth. (Contributed by NM, 5-Aug-1993.)
|- ph   =>    |- ( ps <-> ( ps /\ ph ) )