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**Statement List for Metamath Proof Explorer -** 101-200 - Page 2 of 191
Type	Label	Description
Statement

Theorem	mpcom 101	Modus ponens inference with commutation of antecedents.


Theorem	syl6com 102	Syllogism inference with commuted antecedents.


Theorem	syli 103	Syllogism inference with common nested antecedent.


Theorem	syl6mpi 104	e20 17429 without virtual deductions. (Contributed by Alan Sare, 8-Jul-2011.) (The proof was shortened by Wolf Lammen, 13-Sep-2012.)


Theorem	syl6mpiOLD 105	Obsolete proof of syl6mpi 104.


Theorem	imim12dOLD 106	Deduction combining antecedents and consequents.


Theorem	pm2.43 107	Absorption of redundant antecedent. Also called the "Contraction" or "Hilbert" axiom. Theorem *2.43 of [WhiteheadRussell] p. 106. (The proof was shortened by O'Cat, 15-Aug-2004.)


Theorem	pm2.43i 108	Inference absorbing redundant antecedent. (The proof was shortened by O'Cat, 28-Nov-2008.)


Theorem	pm2.43d 109	Deduction absorbing redundant antecedent. (The proof was shortened by O'Cat, 28-Nov-2008.)


Theorem	pm2.43b 110	Inference absorbing redundant antecedent.


Theorem	syl3c 111	A syllogism inference combined with contraction. e111 17400 without virtual deductions. (Contributed by Alan Sare, 7-Jul-2011.)


Theorem	mpsyl 112	Modus ponens combined with a syllogism inference. (Contributed by Alan Sare, 20-Apr-2011.)


Theorem	pm2.86i 113	Inference based on pm2.86 115. (The proof was shortened by Wolf Lammen, 3-Apr-2013.)


Theorem	pm2.86d 114	Deduction based on pm2.86 115. (The proof was shortened by Wolf Lammen, 3-Apr-2013.)


Theorem	pm2.86 115	Converse of axiom ax-2 5. Theorem *2.86 of [WhiteheadRussell] p. 108. (The proof was shortened by Wolf Lammen, 3-Apr-2013.)


Theorem	pm2.86OLD 116	Obsolete proof of pm2.86 115.


Theorem	pm2.86iOLD 117	Obsolete proof of pm2.86i 113.


Theorem	pm2.86dOLD 118	Obsolete proof of pm2.86d 114.


Theorem	loolinALT 119	The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. This version of loolin 215 does not use ax-3 6, meaning that this theorem is intuitionistically valid. (Contributed by O'Cat, 12-Aug-2004.)


Theorem	loowoz 120	An alternate for the Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz, due to Barbara Wozniakowska, Reports on Mathematical Logic 10, 129-137 (1978). (Contributed by O'Cat, 8-Aug-2004.)


Logical negation

Theorem	con4i 121	Inference rule derived from axiom ax-3 6.


Theorem	con4d 122	Deduction derived from axiom ax-3 6.


Theorem	pm2.21d 123	A contradiction implies anything. Deduction from pm2.21 124.


Theorem	pm2.21 124	From a wff and its negation, anything is true. Theorem *2.21 of [WhiteheadRussell] p. 104. Also called the Duns Scotus law. (The proof was shortened by Wolf Lammen, 14-Sep-2012.)


Theorem	pm2.21OLD 125	Obsolete proof of pm2.21 124.


Theorem	pm2.21i 126	A contradiction implies anything. Inference from pm2.21 124.


Theorem	pm2.24 127	Theorem *2.24 of [WhiteheadRussell] p. 104.


Theorem	pm2.24ii 128	A contradiction implies anything. Inference from pm2.24 127.


Theorem	pm2.18 129	Proof by contradiction. Theorem *2.18 of [WhiteheadRussell] p. 103. Also called the Law of Clavius.


Theorem	notnot2 130	Converse of double negation. Theorem *2.14 of [WhiteheadRussell] p. 102. (The proof was shortened by David Harvey, 5-Sep-1999. An even shorter proof found by Josh Purinton, 29-Dec-2000.)


Theorem	notnotri 131	Inference from double negation.


Theorem	con2d 132	A contraposition deduction.


Theorem	con2 133	Contraposition. Theorem *2.03 of [WhiteheadRussell] p. 100. (The proof was shortened by Wolf Lammen, 12-Feb-2013.)


Theorem	con2OLD 134	Obsolete proof of con2 133.


Theorem	con2dOLD 135	Obsolete proof of con2d 132.


Theorem	con2i 136	A contraposition inference. (The proof was shortened by O'Cat, 28-Nov-2008.)


Theorem	notnot1 137	Converse of double negation. Theorem *2.12 of [WhiteheadRussell] p. 101. (The proof was shortened by Wolf Lammen, 2-Mar-2013.)


Theorem	notnot1OLD 138	Obsolete proof of notnot1 137.


Theorem	notnoti 139	Infer double negation.


Theorem	con1d 140	A contraposition deduction.


Theorem	con1 141	Contraposition. Theorem *2.15 of [WhiteheadRussell] p. 102. (The proof was shortened by Wolf Lammen, 12-Feb-2013.)


Theorem	con1OLD 142	Obsolete proof of con1 141.


Theorem	con1dOLD 143	Obsolete proof of con1d 140.


Theorem	con1i 144	A contraposition inference. (The proof was shortened by O'Cat, 28-Nov-2008.)


Theorem	con3d 145	A contraposition deduction.


Theorem	con3 146	Contraposition. Theorem *2.16 of [WhiteheadRussell] p. 103. (The proof was shortened by Wolf Lammen, 13-Feb-2013.)


Theorem	con3OLD 147	Obsolete proof of con3 146.


Theorem	con3dOLD 148	Obsolete proof of con3d 145.


Theorem	con3i 149	A contraposition inference. (The proof was shortened by Wolf Lammen, 14-Sep-2012.)


Theorem	con3iOLD 150	A contraposition inference.


Theorem	mto 151	The rule of modus tollens.


Theorem	mtod 152	Modus tollens deduction.


Theorem	mtoi 153	Modus tollens inference. (The proof was shortened by Wolf Lammen, 15-Sep-2012.)


Theorem	mtoiOLD 154	Modus tollens inference.


Theorem	mt2 155	A rule similar to modus tollens.


Theorem	mt2d 156	Modus tollens deduction.


Theorem	mt2i 157	Modus tollens inference. (The proof was shortened by Wolf Lammen, 15-Sep-2012.)


Theorem	mt2iOLD 158	Modus tollens inference.


Theorem	mt3 159	A rule similar to modus tollens.


Theorem	mt3d 160	Modus tollens deduction.


Theorem	mt3i 161	Modus tollens inference. (The proof was shortened by Wolf Lammen, 15-Sep-2012.)


Theorem	mt3iOLD 162	Modus tollens inference.


Theorem	mt4d 163	Modus tollens deduction.


Theorem	nsyl3 164	A negated syllogism inference.


Theorem	nsyl 165	A negated syllogism inference. (The proof was shortened by Wolf Lammen, 2-Mar-2013.)


Theorem	nsylOLD 166	Obsolete proof of nsyl 165.


Theorem	nsyld 167	A negated syllogism deduction.


Theorem	nsyl2 168	A negated syllogism inference.


Theorem	nsyli 169	A negated syllogism inference.


Theorem	nsyl4 170	A negated syllogism inference.


Theorem	pm2.24d 171	Deduction version of pm2.24 127.


Theorem	pm2.24i 172	Inference version of pm2.24 127.


Theorem	pm3.2im 173	Theorem *3.2 of [WhiteheadRussell] p. 111, expressed with primitive connectives. (The proof was shortened by Josh Purinton, 29-Dec-2000.)


Theorem	mth8 174	Theorem 8 of [Margaris] p. 60. (The proof was shortened by Josh Purinton, 29-Dec-2000.)


Theorem	jc 175	Inference joining the consequents of two premises.


Theorem	impt 176	Importation theorem expressed with primitive connectives.


Theorem	expt 177	Exportation theorem expressed with primitive connectives.


Theorem	impi 178	An importation inference.


Theorem	expi 179	An exportation inference. (The proof was shortened by O'Cat, 28-Nov-2008.)


Theorem	simprim 180	Simplification. Similar to Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (The proof was shortened by Wolf Lammen, 13-Nov-2012.)


Theorem	simprimOLD 181	Obsolete proof of simprim 180.


Theorem	simplim 182	Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (The proof was shortened by Wolf Lammen, 21-Jul-2012.)


Theorem	simplimOLD 183	Simplification. Similar to Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112.


Theorem	pm2.5 184	Theorem *2.5 of [WhiteheadRussell] p. 107. (The proof was shortened by Wolf Lammen, 9-Oct-2012.)


Theorem	pm2.5OLD 185	obsolete proof of pm2.5 184.


Theorem	pm2.51 186	Theorem *2.51 of [WhiteheadRussell] p. 107.


Theorem	pm2.521 187	Theorem *2.521 of [WhiteheadRussell] p. 107. (The proof was shortened by Wolf Lammen, 8-Oct-2012.)


Theorem	pm2.52 188	Theorem *2.52 of [WhiteheadRussell] p. 107. (The proof was shortened by Wolf Lammen, 8-Oct-2012.)


Theorem	pm2.52OLD 189	Obsolete proof of pm2.52 188.


Theorem	pm2.521OLD 190	Obsolete proof of pm2.521 187.


Theorem	pm2.61OLD 191	Obsolete proof of pm2.61 204.


Theorem	pm2.61i 192	Inference eliminating an antecedent.


Theorem	pm2.61iOLD 193	Obsolete proof of pm2.61i 192.


Theorem	pm2.61d 194	Deduction eliminating an antecedent.


Theorem	pm2.61d1 195	Inference eliminating an antecedent.


Theorem	pm2.61d2 196	Inference eliminating an antecedent.


Theorem	pm2.61ii 197	Inference eliminating two antecedents. (The proof was shortened by Josh Purinton, 29-Dec-2000.)


Theorem	pm2.61nii 198	Inference eliminating two antecedents. (The proof was shortened by Andrew Salmon, 25-May-2011.) (The proof was shortened by Wolf Lammen, 13-Nov-2012.)


Theorem	pm2.61niiOLD 199	Obsolete proof of pm2.61nii 198.


Theorem	pm2.61iii 200	Inference eliminating three antecedents.

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