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
Axiom	ax-ia3 101	'And' introduction. One of the axioms of propositional logic. (Contributed by Mario Carneiro, 31-Jan-2015.)
⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
Theorem	simpl 102	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.)
⊢ ((𝜑 ∧ 𝜓) → 𝜑)
Theorem	simpr 103	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.)
⊢ ((𝜑 ∧ 𝜓) → 𝜓)
Theorem	simpli 104	Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ 𝜑
Theorem	simpld 105	Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	simpri 106	Inference eliminating a conjunct. (Contributed by NM, 15-Jun-1994.)
⊢ (𝜑 ∧ 𝜓)    ⇒   ⊢ 𝜓
Theorem	simprd 107	Deduction eliminating a conjunct. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 3-Oct-2013.)
⊢ (𝜑 → (𝜓 ∧ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	ex 108	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.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	expcom 109	Exportation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ (𝜓 → (𝜑 → 𝜒))
Definition	df-bi 110	This is our first definition, which introduces and defines the biconditional connective ↔. We define a wff of the form (𝜑 ↔ 𝜓) as an abbreviation for ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)). 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 886. 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 887) 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 (𝜑 ↔ 𝜓) (the definiendum i.e. the thing being defined) with ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) (the definiens i.e. the defining expression) in the definition, the definition becomes the previously proved theorem biijust 570. It is impossible to use df-bi 110 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 110 in the proof with the corresponding biijust 570 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 110 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 368 which uses the biconditional instead. Other definitions of the biconditional, such as dfbi3dc 1288, only hold for decidable propositions, not all propositions. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 24-Nov-2017.)
⊢ (((𝜑 ↔ 𝜓) → ((𝜑 → 𝜓) ∧ (𝜓 → 𝜑))) ∧ (((𝜑 → 𝜓) ∧ (𝜓 → 𝜑)) → (𝜑 ↔ 𝜓)))
Theorem	bi1 111	Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Revised by NM, 31-Jan-2015.)
⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
Theorem	bi3 112	Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜑) → (𝜑 ↔ 𝜓)))
Theorem	biimpi 113	Infer an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	sylbi 114	A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	imp 115	Importation inference. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜑 ∧ 𝜓) → 𝜒)
Theorem	impcom 116	Importation inference with commuted antecedents. (Contributed by NM, 25-May-2005.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ ((𝜓 ∧ 𝜑) → 𝜒)
Theorem	impbii 117	Infer an equivalence from an implication and its converse. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	impbidd 118	Deduce an equivalence from two implications. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
Theorem	impbid21d 119	Deduce an equivalence from two implications. (Contributed by Wolf Lammen, 12-May-2013.)
⊢ (𝜓 → (𝜒 → 𝜃))    &   ⊢ (𝜑 → (𝜃 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
Theorem	impbid 120	Deduce an equivalence from two implications. (Contributed by NM, 5-Aug-1993.) (Revised by Wolf Lammen, 3-Nov-2012.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	bi2 121	Property of the biconditional connective. (Contributed by NM, 11-May-1999.) (Proof shortened by Wolf Lammen, 11-Nov-2012.)
⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
Theorem	bicom1 122	Commutative law for equivalence. (Contributed by Wolf Lammen, 10-Nov-2012.)
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
Theorem	bicomi 123	Inference from commutative law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 16-Sep-2013.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜓 ↔ 𝜑)
Theorem	biimpri 124	Infer a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜓 → 𝜑)
Theorem	sylbir 125	A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	pm3.2 126	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.)
⊢ (𝜑 → (𝜓 → (𝜑 ∧ 𝜓)))
Theorem	sylib 127	A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	bicom 128	Commutative law for equivalence. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.) (Revised by NM, 11-Nov-2012.)
⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
Theorem	bicomd 129	Commute two sides of a biconditional in a deduction. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜓))
Theorem	impbid1 130	Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	impbid2 131	Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.) (Proof shortened by Wolf Lammen, 27-Sep-2013.)
⊢ (𝜓 → 𝜒)    &   ⊢ (𝜑 → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	biimpd 132	Deduce an implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	mpbi 133	An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
⊢ 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜓
Theorem	mpbir 134	An inference from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
⊢ 𝜓    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜑
Theorem	mpbid 135	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	mpbii 136	An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
⊢ 𝜓    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylibr 137	A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylibd 138	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	sylbid 139	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	mpbidi 140	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 9-Aug-1994.)
⊢ (𝜃 → (𝜑 → 𝜓))    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜃 → (𝜑 → 𝜒))
Theorem	syl5bi 141	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.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜓 → 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))
Theorem	syl5bir 142	A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 → (𝜓 → 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))
Theorem	syl5ib 143	A mixed syllogism inference. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))
Theorem	syl5ibcom 144	A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜃))
Theorem	syl5ibr 145	A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.) (Revised by NM, 22-Sep-2013.)
⊢ (𝜑 → 𝜃)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜓))
Theorem	syl5ibrcom 146	A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
⊢ (𝜑 → 𝜃)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	biimprd 147	Deduce a converse implication from a logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	biimpcd 148	Deduce a commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜓 → (𝜑 → 𝜒))
Theorem	biimprcd 149	Deduce a converse commuted implication from a logical equivalence. (Contributed by NM, 3-May-1994.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜓))
Theorem	syl6ib 150	A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl6ibr 151	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.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl6bi 152	A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl6bir 153	A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
⊢ (𝜑 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl7bi 154	A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏)))    ⇒   ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏)))
Theorem	syl8ib 155	A syllogism rule of inference. The second premise is used to replace the consequent of the first premise. (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜃 ↔ 𝜏)    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 → 𝜏)))
Theorem	mpbird 156	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	mpbiri 157	An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
⊢ 𝜒    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	sylibrd 158	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	sylbird 159	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	biid 160	Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜑)
Theorem	biidd 161	Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜓))
Theorem	pm5.1im 162	Two propositions are equivalent if they are both true. Closed form of 2th 163. Equivalent to a bi1 111-like version of the xor-connective. This theorem stays true, no matter how you permute its operands. This is evident from its sharper version (𝜑 ↔ (𝜓 ↔ (𝜑 ↔ 𝜓))). (Contributed by Wolf Lammen, 12-May-2013.)
⊢ (𝜑 → (𝜓 → (𝜑 ↔ 𝜓)))
Theorem	2th 163	Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	2thd 164	Two truths are equivalent (deduction rule). (Contributed by NM, 3-Jun-2012.) (Revised by NM, 29-Jan-2013.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	ibi 165	Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 17-Oct-2003.)
⊢ (𝜑 → (𝜑 ↔ 𝜓))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	ibir 166	Inference that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 22-Jul-2004.)
⊢ (𝜑 → (𝜓 ↔ 𝜑))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	ibd 167	Deduction that converts a biconditional implied by one of its arguments, into an implication. (Contributed by NM, 26-Jun-2004.)
⊢ (𝜑 → (𝜓 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	pm5.74 168	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.)
⊢ ((𝜑 → (𝜓 ↔ 𝜒)) ↔ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒)))
Theorem	pm5.74i 169	Distribution of implication over biconditional (inference rule). (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))
Theorem	pm5.74ri 170	Distribution of implication over biconditional (reverse inference rule). (Contributed by NM, 1-Aug-1994.)
⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.74d 171	Distribution of implication over biconditional (deduction rule). (Contributed by NM, 21-Mar-1996.)
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	pm5.74rd 172	Distribution of implication over biconditional (deduction rule). (Contributed by NM, 19-Mar-1997.)
⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
Theorem	bitri 173	An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	bitr2i 174	An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜒 ↔ 𝜑)
Theorem	bitr3i 175	An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	bitr4i 176	An inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	bitrd 177	Deduction form of bitri 173. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	bitr2d 178	Deduction form of bitr2i 174. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜓))
Theorem	bitr3d 179	Deduction form of bitr3i 175. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜃))
Theorem	bitr4d 180	Deduction form of bitr4i 176. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	syl5bb 181	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 ↔ 𝜃))
Theorem	syl5rbb 182	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜃 ↔ 𝜑))
Theorem	syl5bbr 183	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 ↔ 𝜃))
Theorem	syl5rbbr 184	A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜃 ↔ 𝜑))
Theorem	syl6bb 185	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	syl6rbb 186	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜓))
Theorem	syl6bbr 187	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	syl6rbbr 188	A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜓))
Theorem	3imtr3i 189	A mixed syllogism inference, useful for removing a definition from both sides of an implication. (Contributed by NM, 10-Aug-1994.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	3imtr4i 190	A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    &   ⊢ (𝜃 ↔ 𝜓)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	3imtr3d 191	More general version of 3imtr3i 189. Useful for converting conditional definitions in a formula. (Contributed by NM, 8-Apr-1996.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜃 → 𝜏))
Theorem	3imtr4d 192	More general version of 3imtr4i 190. Useful for converting conditional definitions in a formula. (Contributed by NM, 26-Oct-1995.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜓))    &   ⊢ (𝜑 → (𝜏 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜃 → 𝜏))
Theorem	3imtr3g 193	More general version of 3imtr3i 189. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜓 ↔ 𝜃)    &   ⊢ (𝜒 ↔ 𝜏)    ⇒   ⊢ (𝜑 → (𝜃 → 𝜏))
Theorem	3imtr4g 194	More general version of 3imtr4i 190. Useful for converting definitions in a formula. (Contributed by NM, 20-May-1996.) (Proof shortened by Wolf Lammen, 20-Dec-2013.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 ↔ 𝜓)    &   ⊢ (𝜏 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜃 → 𝜏))
Theorem	3bitri 195	A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 ↔ 𝜃)
Theorem	3bitrri 196	A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜃 ↔ 𝜑)
Theorem	3bitr2i 197	A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 ↔ 𝜃)
Theorem	3bitr2ri 198	A chained inference from transitive law for logical equivalence. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜃 ↔ 𝜑)
Theorem	3bitr3i 199	A chained inference from transitive law for logical equivalence. (Contributed by NM, 19-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜒 ↔ 𝜃)
Theorem	3bitr3ri 200	A chained inference from transitive law for logical equivalence. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜃 ↔ 𝜒)