P. 3 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 201-300   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	impbii 201	Infer an equivalence from an implication and its converse. Inference associated with impbi 200. (Contributed by NM, 29-Dec-1992.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 → 𝜑)    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	impbidd 202	Deduce an equivalence from two implications. Double deduction associated with impbi 200 and impbii 201. Deduction associated with impbid 204. (Contributed by Rodolfo Medina, 12-Oct-2010.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜃)))    &   ⊢ (𝜑 → (𝜓 → (𝜃 → 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
Theorem	impbid21d 203	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 204	Deduce an equivalence from two implications. Deduction associated with impbi 200 and impbii 201. (Contributed by NM, 24-Jan-1993.) Revised to prove it from impbid21d 203. (Revised by Wolf Lammen, 3-Nov-2012.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜒 → 𝜓))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	dfbi1 205	Relate the biconditional connective to primitive connectives. See dfbi1ALT 206 for an unusual version proved directly from axioms. (Contributed by NM, 29-Dec-1992.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
Theorem	dfbi1ALT 206	Alternate proof of dfbi1 205. This proof, discovered by Gregory Bush on 8-Mar-2004, has several curious properties. First, it has only 17 steps directly from the axioms and df-bi 199, compared to over 800 steps were the proof of dfbi1 205 expanded into axioms. Second, step 2 demands only the property of "true"; any axiom (or theorem) could be used. It might be thought, therefore, that it is in some sense redundant, but in fact no proof is shorter than this (measured by number of steps). Third, it illustrates how intermediate steps can "blow up" in size even in short proofs. Fourth, the compressed proof is only 182 bytes (or 17 bytes in D-proof notation), but the generated web page is over 200kB with intermediate steps that are essentially incomprehensible to humans (other than Gregory Bush). If there were an obfuscated code contest for proofs, this would be a contender. This "blowing up" and incomprehensibility of the intermediate steps vividly demonstrate the advantages of using many layered intermediate theorems, since each theorem is easier to understand. (Contributed by Gregory Bush, 10-Mar-2004.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) ↔ ¬ ((𝜑 → 𝜓) → ¬ (𝜓 → 𝜑)))
Theorem	biimp 207	Property of the biconditional connective. (Contributed by NM, 11-May-1999.)
⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
Theorem	biimpi 208	Infer an implication from a logical equivalence. Inference associated with biimp 207. (Contributed by NM, 29-Dec-1992.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	sylbi 209	A mixed syllogism inference from a biconditional and an implication. Useful for substituting an antecedent with a definition. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylib 210	A mixed syllogism inference from an implication and a biconditional. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylbb 211	A mixed syllogism inference from two biconditionals. (Contributed by BJ, 30-Mar-2019.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	biimpr 212	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 213	Commutative law for the biconditional. (Contributed by Wolf Lammen, 10-Nov-2012.)
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
Theorem	bicom 214	Commutative law for the biconditional. Theorem *4.21 of [WhiteheadRussell] p. 117. (Contributed by NM, 11-May-1993.)
⊢ ((𝜑 ↔ 𝜓) ↔ (𝜓 ↔ 𝜑))
Theorem	bicomd 215	Commute two sides of a biconditional in a deduction. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜓))
Theorem	bicomi 216	Inference from commutative law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜓 ↔ 𝜑)
Theorem	impbid1 217	Infer an equivalence from two implications. (Contributed by NM, 6-Mar-2007.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜒 → 𝜓)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	impbid2 218	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	impcon4bid 219	A variation on impbid 204 with contraposition. (Contributed by Jeff Hankins, 3-Jul-2009.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (¬ 𝜓 → ¬ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	biimpri 220	Infer a converse implication from a logical equivalence. Inference associated with biimpr 212. (Contributed by NM, 29-Dec-1992.) (Proof shortened by Wolf Lammen, 16-Sep-2013.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (𝜓 → 𝜑)
Theorem	biimpd 221	Deduce an implication from a logical equivalence. Deduction associated with biimp 207 and biimpi 208. (Contributed by NM, 11-Jan-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	mpbi 222	An inference from a biconditional, related to modus ponens. (Contributed by NM, 11-May-1993.)
⊢ 𝜑    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜓
Theorem	mpbir 223	An inference from a biconditional, related to modus ponens. (Contributed by NM, 28-Dec-1992.)
⊢ 𝜓    &   ⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ 𝜑
Theorem	mpbid 224	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	mpbii 225	An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
⊢ 𝜓    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylibr 226	A mixed syllogism inference from an implication and a biconditional. Useful for substituting a consequent with a definition. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylbir 227	A mixed syllogism inference from a biconditional and an implication. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylbbr 228	A mixed syllogism inference from two biconditionals. Note on the various syllogism-like statements in set.mm. The hypothetical syllogism syl 17 infers an implication from two implications (and there are 3syl 18 and 4syl 19 for chaining more inferences). There are four inferences inferring an implication from one implication and one biconditional: sylbi 209, sylib 210, sylbir 227, sylibr 226; four inferences inferring an implication from two biconditionals: sylbb 211, sylbbr 228, sylbb1 229, sylbb2 230; four inferences inferring a biconditional from two biconditionals: bitri 267, bitr2i 268, bitr3i 269, bitr4i 270 (and more for chaining more biconditionals). There are also closed forms and deduction versions of these, like, among many others, syld 47, syl5 34, syl6 35, mpbid 224, bitrd 271, syl5bb 275, syl6bb 279 and variants. (Contributed by BJ, 21-Apr-2019.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜒 → 𝜑)
Theorem	sylbb1 229	A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    ⇒   ⊢ (𝜓 → 𝜒)
Theorem	sylbb2 230	A mixed syllogism inference from two biconditionals. (Contributed by BJ, 21-Apr-2019.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 → 𝜒)
Theorem	sylibd 231	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	sylbid 232	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	mpbidi 233	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 234	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded antecedent with a definition. (Contributed by NM, 12-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜓 → 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))
Theorem	syl5bir 235	A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 → (𝜓 → 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))
Theorem	syl5ib 236	A mixed syllogism inference. (Contributed by NM, 12-Jan-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜃))
Theorem	syl5ibcom 237	A mixed syllogism inference. (Contributed by NM, 19-Jun-2007.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜃))
Theorem	syl5ibr 238	A mixed syllogism inference. (Contributed by NM, 3-Apr-1994.)
⊢ (𝜑 → 𝜃)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 → 𝜓))
Theorem	syl5ibrcom 239	A mixed syllogism inference. (Contributed by NM, 20-Jun-2007.)
⊢ (𝜑 → 𝜃)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	biimprd 240	Deduce a converse implication from a logical equivalence. Deduction associated with biimpr 212 and biimpri 220. (Contributed by NM, 11-Jan-1993.) (Proof shortened by Wolf Lammen, 22-Sep-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜒 → 𝜓))
Theorem	biimpcd 241	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 242	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 243	A mixed syllogism inference from a nested implication and a biconditional. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl6ibr 244	A mixed syllogism inference from a nested implication and a biconditional. Useful for substituting an embedded consequent with a definition. (Contributed by NM, 10-Jan-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜃 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl6bi 245	A mixed syllogism inference. (Contributed by NM, 2-Jan-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl6bir 246	A mixed syllogism inference. (Contributed by NM, 18-May-1994.)
⊢ (𝜑 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜒 → 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	syl7bi 247	A mixed syllogism inference from a doubly nested implication and a biconditional. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜃 → (𝜓 → 𝜏)))    ⇒   ⊢ (𝜒 → (𝜃 → (𝜑 → 𝜏)))
Theorem	syl8ib 248	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 249	A deduction from a biconditional, related to modus ponens. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → 𝜒)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	mpbiri 250	An inference from a nested biconditional, related to modus ponens. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 25-Oct-2012.)
⊢ 𝜒    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → 𝜓)
Theorem	sylibrd 251	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜓 → 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	sylbird 252	A syllogism deduction. (Contributed by NM, 3-Aug-1994.)
⊢ (𝜑 → (𝜒 ↔ 𝜓))    &   ⊢ (𝜑 → (𝜒 → 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜃))
Theorem	biid 253	Principle of identity for logical equivalence. Theorem *4.2 of [WhiteheadRussell] p. 117. This is part of Frege's eighth axiom per Proposition 54 of [Frege1879] p. 50; see also eqid 2778. (Contributed by NM, 2-Jun-1993.)
⊢ (𝜑 ↔ 𝜑)
Theorem	biidd 254	Principle of identity with antecedent. (Contributed by NM, 25-Nov-1995.)
⊢ (𝜑 → (𝜓 ↔ 𝜓))
Theorem	pm5.1im 255	Two propositions are equivalent if they are both true. Closed form of 2th 256. Equivalent to a biimp 207-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 256	Two truths are equivalent. (Contributed by NM, 18-Aug-1993.)
⊢ 𝜑    &   ⊢ 𝜓    ⇒   ⊢ (𝜑 ↔ 𝜓)
Theorem	2thd 257	Two truths are equivalent. Deduction form. (Contributed by NM, 3-Jun-2012.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜑 → 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	monothetic 258	Two self-implications (see id 22) are equivalent. This theorem, rather trivial in our axiomatization, is (the biconditional form of) a standard axiom for monothetic BCI logic. This is the most general theorem of which trujust 1603 is an instance. Relatedly, this would be the justification theorem if the definition of ⊤ were dftru2 1607. (Contributed by BJ, 7-Sep-2022.)
⊢ ((𝜑 → 𝜑) ↔ (𝜓 → 𝜓))
Theorem	ibi 259	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 260	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 261	Deduction that converts a biconditional implied by one of its arguments, into an implication. Deduction associated with ibi 259. (Contributed by NM, 26-Jun-2004.)
⊢ (𝜑 → (𝜓 → (𝜓 ↔ 𝜒)))    ⇒   ⊢ (𝜑 → (𝜓 → 𝜒))
Theorem	pm5.74 262	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 263	Distribution of implication over biconditional (inference form). (Contributed by NM, 1-Aug-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))
Theorem	pm5.74ri 264	Distribution of implication over biconditional (reverse inference form). (Contributed by NM, 1-Aug-1994.)
⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜒))
Theorem	pm5.74d 265	Distribution of implication over biconditional (deduction form). (Contributed by NM, 21-Mar-1996.)
⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))    ⇒   ⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))
Theorem	pm5.74rd 266	Distribution of implication over biconditional (deduction form). (Contributed by NM, 19-Mar-1997.)
⊢ (𝜑 → ((𝜓 → 𝜒) ↔ (𝜓 → 𝜃)))    ⇒   ⊢ (𝜑 → (𝜓 → (𝜒 ↔ 𝜃)))
Theorem	bitri 267	An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.) (Proof shortened by Wolf Lammen, 13-Oct-2012.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	bitr2i 268	An inference from transitive law for logical equivalence. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜒 ↔ 𝜑)
Theorem	bitr3i 269	An inference from transitive law for logical equivalence. (Contributed by NM, 2-Jun-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜓 ↔ 𝜒)    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	bitr4i 270	An inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜓)    ⇒   ⊢ (𝜑 ↔ 𝜒)
Theorem	bitrd 271	Deduction form of bitri 267. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 14-Apr-2013.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	bitr2d 272	Deduction form of bitr2i 268. (Contributed by NM, 9-Jun-2004.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜓))
Theorem	bitr3d 273	Deduction form of bitr3i 269. (Contributed by NM, 14-May-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜑 → (𝜒 ↔ 𝜃))
Theorem	bitr4d 274	Deduction form of bitr4i 270. (Contributed by NM, 30-Jun-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	syl5bb 275	A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 ↔ 𝜃))
Theorem	syl5rbb 276	A syllogism inference from two biconditionals. (Contributed by NM, 1-Aug-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜃 ↔ 𝜑))
Theorem	syl5bbr 277	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜑 ↔ 𝜃))
Theorem	syl5rbbr 278	A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
⊢ (𝜓 ↔ 𝜑)    &   ⊢ (𝜒 → (𝜓 ↔ 𝜃))    ⇒   ⊢ (𝜒 → (𝜃 ↔ 𝜑))
Theorem	syl6bb 279	A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	syl6rbb 280	A syllogism inference from two biconditionals. (Contributed by NM, 5-Aug-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜓))
Theorem	syl6bbr 281	A syllogism inference from two biconditionals. (Contributed by NM, 12-Mar-1993.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜃))
Theorem	syl6rbbr 282	A syllogism inference from two biconditionals. (Contributed by NM, 25-Nov-1994.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜃 ↔ 𝜒)    ⇒   ⊢ (𝜑 → (𝜃 ↔ 𝜓))
Theorem	3imtr3i 283	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 284	A mixed syllogism inference, useful for applying a definition to both sides of an implication. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 → 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    &   ⊢ (𝜃 ↔ 𝜓)    ⇒   ⊢ (𝜒 → 𝜃)
Theorem	3imtr3d 285	More general version of 3imtr3i 283. 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 286	More general version of 3imtr4i 284. 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 287	More general version of 3imtr3i 283. 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 288	More general version of 3imtr4i 284. 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 289	A chained inference from transitive law for logical equivalence. (Contributed by NM, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 ↔ 𝜒)    &   ⊢ (𝜒 ↔ 𝜃)    ⇒   ⊢ (𝜑 ↔ 𝜃)
Theorem	3bitrri 290	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 291	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 292	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 293	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 294	A chained inference from transitive law for logical equivalence. (Contributed by NM, 21-Jun-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜑 ↔ 𝜒)    &   ⊢ (𝜓 ↔ 𝜃)    ⇒   ⊢ (𝜃 ↔ 𝜒)
Theorem	3bitr4i 295	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, 3-Jan-1993.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    &   ⊢ (𝜃 ↔ 𝜓)    ⇒   ⊢ (𝜒 ↔ 𝜃)
Theorem	3bitr4ri 296	A chained inference from transitive law for logical equivalence. (Contributed by NM, 2-Sep-1995.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 ↔ 𝜑)    &   ⊢ (𝜃 ↔ 𝜓)    ⇒   ⊢ (𝜃 ↔ 𝜒)
Theorem	3bitrd 297	Deduction from transitivity of biconditional. (Contributed by NM, 13-Aug-1999.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜏))
Theorem	3bitrrd 298	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜒 ↔ 𝜃))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜏 ↔ 𝜓))
Theorem	3bitr2d 299	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜓 ↔ 𝜏))
Theorem	3bitr2rd 300	Deduction from transitivity of biconditional. (Contributed by NM, 4-Aug-2006.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜒))    &   ⊢ (𝜑 → (𝜃 ↔ 𝜏))    ⇒   ⊢ (𝜑 → (𝜏 ↔ 𝜓))