P. 2 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 101-200   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	pm2.86 101	Converse of Axiom ax-2 7. Theorem *2.86 of [WhiteheadRussell] p. 108. (Contributed by NM, 25-Apr-1994.) (Proof shortened by Wolf Lammen, 3-Apr-2013.)
|- ( ( ( ph -> ps ) -> ( ph -> ch ) ) -> ( ph -> ( ps -> ch ) ) )
Theorem	loolin 102	The Linearity Axiom of the infinite-valued sentential logic (L-infinity) of Lukasiewicz. (Contributed by O'Cat, 12-Aug-2004.)
|- ( ( ( ph -> ps ) -> ( ps -> ph ) ) -> ( ps -> ph ) )
Theorem	loowoz 103	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.)
|- ( ( ( ph -> ps ) -> ( ph -> ch ) ) -> ( ( ps -> ph ) -> ( ps -> ch ) ) )
1.2.4  Logical conjunction and logical equivalence
Syntax	wa 104	Extend wff definition to include conjunction ('and').
wff ( ph /\ ps )
Syntax	wb 105	Extend our wff definition to include the biconditional connective.
wff ( ph <-> ps )
Axiom	ax-ia1 106	Left 'and' elimination. One of the axioms of propositional logic. Use its alias simpl 109 instead for naming consistency with set.mm. (New usage is discouraged.) (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( ( ph /\ ps ) -> ph )
Axiom	ax-ia2 107	Right 'and' elimination. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.) Use its alias simpr 110 instead for naming consistency with set.mm. (New usage is discouraged.)
|- ( ( ph /\ ps ) -> ps )
Axiom	ax-ia3 108	'And' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
|- ( ph -> ( ps -> ( ph /\ ps ) ) )
Theorem	simpl 109	Elimination of a conjunct. Theorem *3.26 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
|- ( ( ph /\ ps ) -> ph )
Theorem	simpr 110	Elimination of a conjunct. Theorem *3.27 (Simp) of [WhiteheadRussell] p. 112. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Nov-2012.)
|- ( ( ph /\ ps ) -> ps )
Theorem	simpli 111	Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
|- ( ph /\ ps )   =>    |- ph
Theorem	simpld 112	Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps /\ ch ) )   =>    |- ( ph -> ps )
Theorem	simpri 113	Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
|- ( ph /\ ps )   =>    |- ps
Theorem	simprd 114	Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
|- ( ph -> ( ps /\ ch ) )   =>    |- ( ph -> ch )
Theorem	ex 115	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.) (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ph -> ( ps -> ch ) )
Theorem	expcom 116	Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
|- ( ( ph /\ ps ) -> ch )   =>    |- ( ps -> ( ph -> ch ) )
Definition	df-bi 117	This is our first definition, which introduces and defines the biconditional connective <->, also called biimplication. We define a wff of the form ( ph <-> ps ) as an abbreviation for ( ( ph -> ps ) /\ ( ps -> ph ) ). Unlike most traditional developments, we have chosen not to have a separate symbol such as "Df." to mean "is defined as." Instead, we will later use the biconditional connective for this purpose, as it allows us to use logic to manipulate definitions directly. For an example of such a definition, see df-3or 979. This greatly simplifies many proofs since it eliminates the need for a separate mechanism for introducing and eliminating definitions. Of course, we cannot use this mechanism to define the biconditional itself, since it hasn't been introduced yet. Instead, we use a more general form of definition, described as follows. In its most general form, a definition is simply an assertion that introduces a new symbol (or a new combination of existing symbols, as in df-3an 980) that is eliminable and does not strengthen the existing language. The latter requirement means that the set of provable statements not containing the new symbol (or new combination) should remain exactly the same after the definition is introduced. Our definition of the biconditional may look unusual compared to most definitions, but it strictly satisfies these requirements. The justification for our definition is that if we mechanically replace ( ph <-> ps ) (the definiendum i.e. the thing being defined) with ( ( ph -> ps ) /\ ( ps -> ph ) ) (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem biijust 641. It is impossible to use df-bi 117 to prove any statement expressed in the original language that can't be proved from the original axioms, because if we simply replace each instance of df-bi 117 in the proof with the corresponding biijust 641 instance, we will end up with a proof from the original axioms. Note that from Metamath's point of view, a definition is just another axiom - i.e. an assertion we claim to be true - but from our high level point of view, we are are not strengthening the language. To indicate this fact, we prefix definition labels with "df-" instead of "ax-". (This prefixing is an informal convention that means nothing to the Metamath proof verifier; it is just for human readability.) df-bi 117 itself is a conjunction of two implications (to avoid using the biconditional in its own definition), but once we have the biconditional, we can prove dfbi2 388 which uses the biconditional instead. Other definitions of the biconditional, such as dfbi3dc 1397, only hold for decidable propositions, not all propositions. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 24-Nov-2017.)
|- ( ( ( ph <-> ps ) -> ( ( ph -> ps ) /\ ( ps -> ph ) ) ) /\ ( ( ( ph -> ps ) /\ ( ps -> ph ) ) -> ( ph <-> ps ) ) )
Theorem	biimp 118	Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Revised by NM, 31-Jan-2015.)
|- ( ( ph <-> ps ) -> ( ph -> ps ) )
Theorem	bi3 119	Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
|- ( ( ph -> ps ) -> ( ( ps -> ph ) -> ( ph <-> ps ) ) )
Theorem	biimpi 120	Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   =>    |- ( ph -> ps )
Theorem	sylbi 121	A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 3-Jan-1993.)
|- ( ph <-> ps )   &    |- ( ps -> ch )   =>    |- ( ph -> ch )
Theorem	sylib 122	A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.)
|- ( ph -> ps )   &    |- ( ps <-> ch )   =>    |- ( ph -> ch )
Theorem	sylbb 123	A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.)
|- ( ph <-> ps )   &    |- ( ps <-> ch )   =>    |- ( ph -> ch )
Theorem	imp 124	Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ( ph /\ ps ) -> ch )
Theorem	impcom 125	Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
|- ( ph -> ( ps -> ch ) )   =>    |- ( ( ps /\ ph ) -> ch )
Theorem	impbii 126	Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   &    |- ( ps -> ph )   =>    |- ( ph <-> ps )
Theorem	impbidd 127	Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   &    |- ( ph -> ( ps -> ( th -> ch ) ) )   =>    |- ( ph -> ( ps -> ( ch <-> th ) ) )
Theorem	impbid21d 128	Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)
|- ( ps -> ( ch -> th ) )   &    |- ( ph -> ( th -> ch ) )   =>    |- ( ph -> ( ps -> ( ch <-> th ) ) )
Theorem	impbid 129	Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( ch -> ps ) )   =>    |- ( ph -> ( ps <-> ch ) )
Theorem	biimpr 130	Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
|- ( ( ph <-> ps ) -> ( ps -> ph ) )
Theorem	bicom1 131	Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.)
|- ( ( ph <-> ps ) -> ( ps <-> ph ) )
Theorem	bicomi 132	Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.)
|- ( ph <-> ps )   =>    |- ( ps <-> ph )
Theorem	biimpri 133	Infer a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)
|- ( ph <-> ps )   =>    |- ( ps -> ph )
Theorem	sylibr 134	A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   &    |- ( ch <-> ps )   =>    |- ( ph -> ch )
Theorem	sylbir 135	A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.)
|- ( ps <-> ph )   &    |- ( ps -> ch )   =>    |- ( ph -> ch )
Theorem	sylbbr 136	A mixed syllogism inference from two biconditionals. Note on the various syllogism-like statements in set.mm. The hypothetical syllogism syl 14 infers an implication from two implications (and there are 3syl 17 and 4syl 18 for chaining more inferences). There are four inferences inferring an implication from one implication and one biconditional: sylbi 121, sylib 122, sylbir 135, sylibr 134; four inferences inferring an implication from two biconditionals: sylbb 123, sylbbr 136, sylbb1 137, sylbb2 138; four inferences inferring a biconditional from two biconditionals: bitri 184, bitr2i 185, bitr3i 186, bitr4i 187 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 45, syl5 32, syl6 33, mpbid 147, bitrd 188, bitrid 192, bitrdi 196 and variants. (Contributed by BJ, 21-Apr-2019.)
|- ( ph <-> ps )   &    |- ( ps <-> ch )   =>    |- ( ch -> ph )
Theorem	sylbb1 137	A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
|- ( ph <-> ps )   &    |- ( ph <-> ch )   =>    |- ( ps -> ch )
Theorem	sylbb2 138	A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
|- ( ph <-> ps )   &    |- ( ch <-> ps )   =>    |- ( ph -> ch )
Theorem	pm3.2 139	Join antecedents with conjunction. Theorem *3.2 of [WhiteheadRussell] p. 111. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2012.) (Proof shortened by Jia Ming, 17-Nov-2020.)
|- ( ph -> ( ps -> ( ph /\ ps ) ) )
Theorem	bicom 140	Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 11-Nov-2012.)
|- ( ( ph <-> ps ) <-> ( ps <-> ph ) )
Theorem	bicomd 141	Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ch <-> ps ) )
Theorem	impbid1 142	Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ch -> ps )   =>    |- ( ph -> ( ps <-> ch ) )
Theorem	impbid2 143	Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
|- ( ps -> ch )   &    |- ( ph -> ( ch -> ps ) )   =>    |- ( ph -> ( ps <-> ch ) )
Theorem	biimpd 144	Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ps -> ch ) )
Theorem	mpbi 145	An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
|- ph   &    |- ( ph <-> ps )   =>    |- ps
Theorem	mpbir 146	An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
|- ps   &    |- ( ph <-> ps )   =>    |- ph
Theorem	mpbid 147	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ps )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ch )
Theorem	mpbii 148	An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
|- ps   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ch )
Theorem	sylibd 149	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( ch <-> th ) )   =>    |- ( ph -> ( ps -> th ) )
Theorem	sylbid 150	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( ch -> th ) )   =>    |- ( ph -> ( ps -> th ) )
Theorem	mpbidi 151	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
|- ( th -> ( ph -> ps ) )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( th -> ( ph -> ch ) )
Theorem	biimtrid 152	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch -> ( ps -> th ) )   =>    |- ( ch -> ( ph -> th ) )
Theorem	biimtrrid 153	A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
|- ( ps <-> ph )   &    |- ( ch -> ( ps -> th ) )   =>    |- ( ch -> ( ph -> th ) )
Theorem	imbitrid 154	A mixed syllogism inference. (Contributed by NM, 12-Jan-1993.)
|- ( ph -> ps )   &    |- ( ch -> ( ps <-> th ) )   =>    |- ( ch -> ( ph -> th ) )
Theorem	syl5ibcom 155	A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
|- ( ph -> ps )   &    |- ( ch -> ( ps <-> th ) )   =>    |- ( ph -> ( ch -> th ) )
Theorem	imbitrrid 156	A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised by NM, 22-Sep-2013.)
|- ( ph -> th )   &    |- ( ch -> ( ps <-> th ) )   =>    |- ( ch -> ( ph -> ps ) )
Theorem	syl5ibrcom 157	A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
|- ( ph -> th )   &    |- ( ch -> ( ps <-> th ) )   =>    |- ( ph -> ( ch -> ps ) )
Theorem	biimprd 158	Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ( ch -> ps ) )
Theorem	biimpcd 159	Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ps -> ( ph -> ch ) )
Theorem	biimprcd 160	Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ch -> ( ph -> ps ) )
Theorem	imbitrdi 161	A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ch <-> th )   =>    |- ( ph -> ( ps -> th ) )
Theorem	syl6ibr 162	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps -> ch ) )   &    |- ( th <-> ch )   =>    |- ( ph -> ( ps -> th ) )
Theorem	syl6bi 163	A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ch -> th )   =>    |- ( ph -> ( ps -> th ) )
Theorem	syl6bir 164	A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
|- ( ph -> ( ch <-> ps ) )   &    |- ( ch -> th )   =>    |- ( ph -> ( ps -> th ) )
Theorem	syl7bi 165	A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch -> ( th -> ( ps -> ta ) ) )   =>    |- ( ch -> ( th -> ( ph -> ta ) ) )
Theorem	syl8ib 166	A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.)
|- ( ph -> ( ps -> ( ch -> th ) ) )   &    |- ( th <-> ta )   =>    |- ( ph -> ( ps -> ( ch -> ta ) ) )
Theorem	mpbird 167	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ch )   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ps )
Theorem	mpbiri 168	An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
|- ch   &    |- ( ph -> ( ps <-> ch ) )   =>    |- ( ph -> ps )
Theorem	sylibrd 169	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
|- ( ph -> ( ps -> ch ) )   &    |- ( ph -> ( th <-> ch ) )   =>    |- ( ph -> ( ps -> th ) )
Theorem	sylbird 170	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
|- ( ph -> ( ch <-> ps ) )   &    |- ( ph -> ( ch -> th ) )   =>    |- ( ph -> ( ps -> th ) )
Theorem	biid 171	Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ph )
Theorem	biidd 172	Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
|- ( ph -> ( ps <-> ps ) )
Theorem	pm5.1im 173	Two propositions are equivalent if they are both true. Closed form of 2th 174. Equivalent to a biimp 118-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version ( ph <-> ( ps <-> ( ph <-> ps ) ) ). (Contributed by Wolf Lammen, 12-May-2013.)
|- ( ph -> ( ps -> ( ph <-> ps ) ) )
Theorem	2th 174	Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
|- ph   &    |- ps   =>    |- ( ph <-> ps )
Theorem	2thd 175	Two truths are equivalent (deduction form). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.)
|- ( ph -> ps )   &    |- ( ph -> ch )   =>    |- ( ph -> ( ps <-> ch ) )
Theorem	ibi 176	Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)
|- ( ph -> ( ph <-> ps ) )   =>    |- ( ph -> ps )
Theorem	ibir 177	Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)
|- ( ph -> ( ps <-> ph ) )   =>    |- ( ph -> ps )
Theorem	ibd 178	Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.)
|- ( ph -> ( ps -> ( ps <-> ch ) ) )   =>    |- ( ph -> ( ps -> ch ) )
Theorem	pm5.74 179	Distribution of implication over biconditional. Theorem *5.74 of [WhiteheadRussell] p. 126. (Contributed by NM, 1-Aug-1994.) (Proof shortened by Wolf Lammen, 11-Apr-2013.)
|- ( ( ph -> ( ps <-> ch ) ) <-> ( ( ph -> ps ) <-> ( ph -> ch ) ) )
Theorem	pm5.74i 180	Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
|- ( ph -> ( ps <-> ch ) )   =>    |- ( ( ph -> ps ) <-> ( ph -> ch ) )
Theorem	pm5.74ri 181	Distribution of implication over biconditional (reverse inference form). (Contributed by NM, 1-Aug-1994.)
|- ( ( ph -> ps ) <-> ( ph -> ch ) )   =>    |- ( ph -> ( ps <-> ch ) )
Theorem	pm5.74d 182	Distribution of implication over biconditional (deduction form). (Contributed by NM, 21-Mar-1996.)
|- ( ph -> ( ps -> ( ch <-> th ) ) )   =>    |- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) )
Theorem	pm5.74rd 183	Distribution of implication over biconditional (deduction form). (Contributed by NM, 19-Mar-1997.)
|- ( ph -> ( ( ps -> ch ) <-> ( ps -> th ) ) )   =>    |- ( ph -> ( ps -> ( ch <-> th ) ) )
Theorem	bitri 184	An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
|- ( ph <-> ps )   &    |- ( ps <-> ch )   =>    |- ( ph <-> ch )
Theorem	bitr2i 185	An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ps <-> ch )   =>    |- ( ch <-> ph )
Theorem	bitr3i 186	An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ps <-> ph )   &    |- ( ps <-> ch )   =>    |- ( ph <-> ch )
Theorem	bitr4i 187	An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch <-> ps )   =>    |- ( ph <-> ch )
Theorem	bitrd 188	Deduction form of bitri 184. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( ch <-> th ) )   =>    |- ( ph -> ( ps <-> th ) )
Theorem	bitr2d 189	Deduction form of bitr2i 185. (Contributed by NM, 9-Jun-2004.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( ch <-> th ) )   =>    |- ( ph -> ( th <-> ps ) )
Theorem	bitr3d 190	Deduction form of bitr3i 186. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( ps <-> th ) )   =>    |- ( ph -> ( ch <-> th ) )
Theorem	bitr4d 191	Deduction form of bitr4i 187. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ph -> ( th <-> ch ) )   =>    |- ( ph -> ( ps <-> th ) )
Theorem	bitrid 192	A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
|- ( ph <-> ps )   &    |- ( ch -> ( ps <-> th ) )   =>    |- ( ch -> ( ph <-> th ) )
Theorem	bitr2id 193	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
|- ( ph <-> ps )   &    |- ( ch -> ( ps <-> th ) )   =>    |- ( ch -> ( th <-> ph ) )
Theorem	bitr3id 194	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
|- ( ps <-> ph )   &    |- ( ch -> ( ps <-> th ) )   =>    |- ( ch -> ( ph <-> th ) )
Theorem	bitr3di 195	A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ps <-> th )   =>    |- ( ph -> ( ch <-> th ) )
Theorem	bitrdi 196	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ch <-> th )   =>    |- ( ph -> ( ps <-> th ) )
Theorem	bitr2di 197	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( ch <-> th )   =>    |- ( ph -> ( th <-> ps ) )
Theorem	bitr4di 198	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
|- ( ph -> ( ps <-> ch ) )   &    |- ( th <-> ch )   =>    |- ( ph -> ( ps <-> th ) )
Theorem	bitr4id 199	A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
|- ( ps <-> ch )   &    |- ( ph -> ( th <-> ch ) )   =>    |- ( ph -> ( ps <-> th ) )
Theorem	3imtr3i 200	A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)
|- ( ph -> ps )   &    |- ( ph <-> ch )   &    |- ( ps <-> th )   =>    |- ( ch -> th )