mmtheorems14 - Metamath Proof Explorer

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**Statement List for Metamath Proof Explorer -** 1301-1400 - Page 14 of 105
Type	Label	Description
Statement

Theorem	cbvex2v 1301	Rule used to change bound variables with implicit substitution.
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Theorem	cbvald 1302	Deduction used to change bound variables with implicit substitution, particularly useful in conjunction with dvelim 1334.


Theorem	cbvexd 1303	Deduction used to change bound variables with implicit substitution, particularly useful in conjunction with dvelim 1334.


Theorem	cbvex4v 1304	Rule used to change bound variables with implicit substitution.
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Theorem	eeanv 1305	Rearrange existential quantifiers.
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Theorem	eeeanv 1306	Rearrange existential quantifiers.
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Theorem	ee4anv 1307	Rearrange existential quantifiers.
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Theorem	nexdv 1308	Deduction for generalization rule for negated wff.


Theorem	chvarv 1309	Implicit substitution of for into a theorem.


Theorem	cleljust 1310	When the class variables in definition df-clel 1449 are replaced with set variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the set variables in wel 1106 with the class variables in wcel 1105.
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More substitution theorems

Theorem	equsb3lem 1311	Lemma for equsb3 1312.

Theorem	equsb3 1312	Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)


Theorem	elsb3 1313	Substitution applied to an atomic membership wff.


Theorem	hbs1 1314	is not free in when and are distinct.


Theorem	hbsb 1315	If is not free in , it is not free in when and are distinct.


Theorem	sbcom2 1316	Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint).


Theorem	2sb5 1317	Equivalence for double substitution.
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Theorem	2sb6 1318	Equivalence for double substitution.
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Theorem	sb6a 1319	Equivalence for substitution.


Theorem	2sb5rf 1320	Reversed double substitution.
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Theorem	2sb6rf 1321	Reversed double substitution.
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Theorem	sb7 1322	An alternate definition of proper substitution df-sb 1155. By introducing a dummy variable in the definiens, we are able to eliminate any distinct variable restrictions among the variables , , and of the definiendum. No distinct variable conflicts arise because effectively insulates from . To achieve this, we use a chain of two substitutions in the form of sb5 1252, first for then for . Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 1441. Theorem sb7f 1323 provides a version where and don't have to be distinct.
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Theorem	sb7f 1323	This version of sb7 1322 does not require that and be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 1190 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1155 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.)
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Theorem	sb10f 1324	Hao Wang's identity axiom P6 in Irving Copi, Symbolic Logic (5th ed., 1979), p. 328. In traditional predicate calculus, this is a sole axiom for identity from which the usual ones can be derived.
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Theorem	sbid2v 1325	An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint).


Theorem	sbelx 1326	Elimination of substitution.
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Theorem	sbel2x 1327	Elimination of double substitution.
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Theorem	sbal1 1328	A theorem used in elimination of disjoint variable restriction on and by replacing it with a distinctor .


Theorem	sbal 1329	Move universal quantifier in and out of substitution.


Theorem	sbex 1330	Move existential quantifier in and out of substitution.


Theorem	sbalv 1331	Quantify with new variable inside substitution.


Theorem	exsb 1332	An equivalent expression for existence.


Theorem	2exsb 1333	An equivalent expression for double existence.
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Theorem	dvelim 1334	This theorem can be used to eliminate a distinct variable restriction on and and replace it with the "distinctor" as an antecedent. is considered to contain , and to have replaced with , and we don't require that and be distinct (if they aren't, the distinctor will become false and "protect" the consequent). To obtain a closed theorem form of this inference, conjoin the hypotheses and apply dvelimdf 1235.


Theorem	dveeq1 1335	Quantifier introduction when one pair of variables is distinct.


Theorem	dveel1 1336	Quantifier introduction when one pair of variables is distinct.


Theorem	dveel2 1337	Quantifier introduction when one pair of variables is distinct.


Theorem	sbal2 1338	Move quantifier in and out of substitution.


Theorem	ax15 1339	Axiom ax-15 1109 is redundant if we assume ax-17 1190. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'. Note that is a dummy variable introduced in the proof. On the web page, it is implicitly assumed to be distinct from all other variables. (This is made explicit in the database file set.mm). Its purpose is to satisfy the distinct variable requirements of dveel2 1337 and dvelim 1334. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements.


Theorem	ax11eq 1340	Basis step for constructing a substitution instance of ax-11o 1202 without using ax-11o 1202. Atomic formula for equality predicate.


Theorem	ax11el 1341	Basis step for constructing a substitution instance of ax-11o 1202 without using ax-11o 1202. Atomic formula for membership predicate.


Theorem	ax11f 1342	Basis step for constructing a substitution instance of ax-11o 1202 without using ax-11o 1202. We can start with any formula in which is not free.


Theorem	ax11indn 1343	Induction step for constructing a substitution instance of ax-11o 1202 without using ax-11o 1202. Negation case.


Theorem	ax11indi 1344	Induction step for constructing a substitution instance of ax-11o 1202 without using ax-11o 1202. Implication case.


Theorem	ax11indalem 1345	Lemma for ax11inda2 1347 and ax11inda 1348.

Theorem	ax11inda2ALT 1346	A proof of ax11inda2 1347 that is slightly more direct.


Theorem	ax11inda2 1347	Induction step for constructing a substitution instance of ax-11o 1202 without using ax-11o 1202. Quantification case. When and are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 1348.


Theorem	ax11inda 1348	Induction step for constructing a substitution instance of ax-11o 1202 without using ax-11o 1202. Quantification case. (When and are distinct, ax11inda2 1347 may be used instead to avoid the dummy variable in the proof.)