mmtheorems13 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1201-1300 - Page 13 of 123

Type	Label	Description
Statement
Theorem	cbv3 1201	Rule used to change bound variables, using implicit substitition.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph -> ps))   =>   |- (A.xph -> A.yps)
Theorem	cbval 1202	Rule used to change bound variables, using implicit substitition.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (A.xph <-> A.yps)
Theorem	cbvex 1203	Rule used to change bound variables, using implicit substitition.
|- (ph -> A.yph)   &   |- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- (E.xph <-> E.yps)
Theorem	chvar 1204	Implicit substitution of y for x into a theorem. (Contributed by Raph Levien, 9-Jul-2003.)
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   &   |- ph   =>   |- ps
Theorem	equvini 1205	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109, however we do not require z to be distinct from x and y (making the proof longer).
|- (x = y -> E.z(x = z /\ z = y))
Theorem	hbequid 1206	Bound-variable hypothesis builder for x = x. This theorem tells us that x is effectively not free in x = x, even though it is technically free according to the traditional definition of free variable. (The proof shows that this can be proved without ax-9 1001, even though the theorem equid 1162 cannot be. A shorter proof that uses ax-9 1001 is obtainable from equid 1162 and hbth 1037.)
|- (x = x -> A.x x = x)
Substitution (without distinct variables)
Syntax	wsbc 1207	Extend wff notation to include the proper substitution of a class for a set. Read this notation as "the proper substitution of class A for set variable x in wff ph." (The purpose of introducing wff [A / x]ph here is to allow us to express i.e. "prove" the wsb 1208 of predicate calculus in terms of the wsbc 1207 of set theory, so that we don't "overload" its connectives with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variable A is introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-sbc 1987 for more information on the set theory usage of wsbc 1207.)
wff [A / x]ph
Theorem	wsb 1208	Extend wff definition to include proper substitution (read "the wff that results when y is properly substituted for x in wff ph"). (Instead of introducing wsb 1208 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wsbc 1207. This lets us avoid overloading its connectives, thus preventing ambiguity that would complicate some Metamath parsers. Note: To see the proof steps of this syntax proof, type "show proof wsb /all" in the Metamath program.)
wff [y / x]ph
Definition	df-sb 1209	Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use [y / x]ph to mean "the wff that results when y is properly substituted for x in the wff ph." We can also use [y / x]ph in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 1222. Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "ph(y) is the wff that results when y is properly substituted for x in ph(x)." For example, if the original ph(x) is x = y, then ph(y) is y = y, from which we obtain that ph(x) is x = x. So what exactly does ph(x) mean? Curry's notation solves this problem. In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a remarkable little formula that is exactly equivalent and gives us a single direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 1266, sbcom2 1373 and sbid2v 1382). Note that our definition is valid even when x and y are replaced with the same variable, as sbid 1221 shows. We achieve this by having x free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 1379 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 1263. When x and y are distinct, we can express proper substitution with the simpler expressions of sb5 1306 and sb6 1305. There are no restrictions on any of the variables, including what variables may occur in wff ph.
|- ([y / x]ph <-> ((x = y -> ph) /\ E.x(x = y /\ ph)))
Theorem	sbimi 1210	Infer substitution into antecedent and consequent of an implication.
|- (ph -> ps)   =>   |- ([y / x]ph -> [y / x]ps)
Theorem	sbbii 1211	Infer substitution into both sides of a logical equivalence.
|- (ph <-> ps)   =>   |- ([y / x]ph <-> [y / x]ps)
Theorem	drsb1 1212	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|- (A.x x = y -> ([z / x]ph <-> [z / y]ph))
Theorem	sb1 1213	One direction of a simplified definition of substitution.
|- ([y / x]ph -> E.x(x = y /\ ph))
Theorem	sb2 1214	One direction of a simplified definition of substitution.
|- (A.x(x = y -> ph) -> [y / x]ph)
Theorem	sbequ1 1215	An equality theorem for substitution.
|- (x = y -> (ph -> [y / x]ph))
Theorem	sbequ2 1216	An equality theorem for substitution.
|- (x = y -> ([y / x]ph -> ph))
Theorem	stdpc7 1217	One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1164.) Translated to traditional notation, it can be read: "x = y -> (ph(x, x) -> ph(x, y)), provided that y is free for x in ph(x, y)." Axiom 7 of [Mendelson] p. 95.
|- (x = y -> ([x / y]ph -> ph))
Theorem	sbequ12 1218	An equality theorem for substitution.
|- (x = y -> (ph <-> [y / x]ph))
Theorem	sbequ12r 1219	An equality theorem for substitution.
|- (x = y -> ([x / y]ph <-> ph))
Theorem	sbequ12a 1220	An equality theorem for substitution.
|- (x = y -> ([y / x]ph <-> [x / y]ph))
Theorem	sbid 1221	An identity theorem for substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint).
|- ([x / x]ph <-> ph)
Theorem	stdpc4 1222	The specialization axiom of standard predicate calculus. It states that if a statement ph holds for all x, then it also holds for the specific case of y (properly) substituted for x. Translated to traditional notation, it can be read: "A.xph(x) -> ph(y), provided that y is free for x in ph(x)." Axiom 4 of [Mendelson] p. 69. See also a4sbc 1990 and ra4sbc 2047.
|- (A.xph -> [y / x]ph)
Theorem	sbf 1223	Substitution for a variable not free in a wff does not affect it.
|- (ph -> A.xph)   =>   |- ([y / x]ph <-> ph)
Theorem	sbf2 1224	Substitution has no effect on a bound variable.
|- ([y / x]A.xph <-> A.xph)
Theorem	sb6x 1225	Equivalence involving substitution for a variable not free.
|- (ph -> A.xph)   =>   |- ([y / x]ph <-> A.x(x = y -> ph))
Theorem	hbs1f 1226	If x is not free in ph, it is not free in [y / x]ph.
|- (ph -> A.xph)   =>   |- ([y / x]ph -> A.x[y / x]ph)
Theorem	sbequ5 1227	Substitution does not change an identical variable specifier.
|- ([w / z]A.x x = y <-> A.x x = y)
Theorem	sbequ6 1228	Substitution does not change a distinctor.
|- ([w / z] -. A.x x = y <-> -. A.x x = y)
Theorem	sbt 1229	A substitution into a theorem remains true. (See chvar 1204 and chvarv 1365 for versions, using implicit substitition.
|- ph   =>   |- [y / x]ph
Theorem	equsb1 1230	Substitution applied to an atomic wff.
|- [y / x]x = y
Theorem	equsb2 1231	Substitution applied to an atomic wff.
|- [y / x]y = x
Theorem	sbied 1232	Conversion of implicit substitution to explicit substitution (deduction version of sbie 1233).
|- (ph -> A.xph)   &   |- (ph -> (ch -> A.xch))   &   |- (ph -> (x = y -> (ps <-> ch)))   =>   |- (ph -> ([y / x]ps <-> ch))
Theorem	sbie 1233	Conversion of implicit substitution to explicit substitution.
|- (ps -> A.xps)   &   |- (x = y -> (ph <-> ps))   =>   |- ([y / x]ph <-> ps)
Theorems using axiom ax-11
Theorem	equs5a 1234	A property related to substitution that unlike equs5 1258 doesn't require a distinctor antecedent.
|- (E.x(x = y /\ A.yph) -> A.x(x = y -> ph))
Theorem	equs5e 1235	A property related to substitution that unlike equs5 1258 doesn't require a distinctor antecedent.
|- (E.x(x = y /\ ph) -> A.x(x = y -> E.yph))
Theorem	sb4a 1236	A version of sb4 1260 that doesn't require a distinctor antecedent.
|- ([y / x]A.yph -> A.x(x = y -> ph))
Theorem	equs45f 1237	Two ways of expressing substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))
Theorem	sb6f 1238	Equivalence for substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph <-> A.x(x = y -> ph))
Theorem	sb5f 1239	Equivalence for substitution when y is not free in ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph <-> E.x(x = y /\ ph))
Theorem	sb4e 1240	One direction of a simplified definition of substitution that unlike sb4 1260 doesn't require a distinctor antecedent.
|- ([y / x]ph -> A.x(x = y -> E.yph))
Theorem	hbsb2a 1241	Special case of a bound-variable hypothesis builder for substitution.
|- ([y / x]A.yph -> A.x[y / x]ph)
Theorem	hbsb2e 1242	Special case of a bound-variable hypothesis builder for substitution.
|- ([y / x]ph -> A.x[y / x]E.yph)
Theorem	hbsb3 1243	If y is not free in ph, x is not free in [y / x]ph.
|- (ph -> A.yph)   =>   |- ([y / x]ph -> A.x[y / x]ph)
Predicate calculus with distinct variables
The axiom of quantifier introduction ax-17
Theorem	a4imv 1244	A version of a4im 1196 with a distinct variable requirement instead of a bound variable hypothesis.
|- (x = y -> (ph -> ps))   =>   |- (A.xph -> ps)
Theorem	aev 1245	A "distinctor elimination" lemma with no restrictions on variables in the consequent, proved without using ax-16 1247. The proof is unusual in that it involves linking 17 implications, which might provide an interesting challenge for an automated theorem prover.
|- (A.x x = y -> A.z w = v)
Derive the axiom of distinct variables ax-16
Theorem	ax16 1246	Theorem showing that ax-16 1247 is redundant if ax-17 1007 is included in the axiom system. The important part of the proof is provided by aev 1245. See ax16ALT 1309 for an alternate proof that does not require ax-10 1002 or ax-12 1004. This theorem should not be referenced in any proof. Instead, use ax-16 1247 below so that theorems needing ax-16 1247 can be more easily identified.
|- (A.x x = y -> (ph -> A.xph))
Axiom	ax-16 1247	Axiom of Distinct Variables. The only axiom of predicate calculus requiring that variables be distinct (if we consider ax-17 1007 to be a metatheorem and not an axiom). Axiom scheme C16' in [Megill] p. 448 (p. 16 of the preprint). It apparently does not otherwise appear in the literature but is easily proved from textbook predicate calculus by cases. It is a somewhat bizarre axiom since the antecedent is always false in set theory (see dtru 2831), but nonetheless it is technically necessary as you can see from its uses. This axiom is redundant if we include ax-17 1007; see theorem ax16 1246. Alternately, ax-17 1007 becomes logically redundant in the presence of this axiom, but without ax-17 1007 we lose the more powerful metalogic that results from being able to express the concept of a set variable not occurring in a wff (as opposed to just two set variables being distinct). We retain ax-16 1247 here to provide logical completeness for systems with the simpler metalogic that results from omitting ax-17 1007, which might be easier to study for some theoretical purposes.
|- (A.x x = y -> (ph -> A.xph))
Theorem	ax17eq 1248	Theorem to add distinct quantifier to atomic formula. (This theorem demonstrates the induction basis for ax-17 1007 considered as a metatheorem. Do not use it for later proofs - use ax-17 1007 instead, to avoid reference to the redundant axiom ax-16 1247.)
|- (x = y -> A.z x = y)
Theorem	dveeq2 1249	Quantifier introduction when one pair of variables is distinct.
|- (-. A.x x = y -> (z = y -> A.x z = y))
Theorem	dveeq2ALT 1250	Version of dveeq2 1249 using ax-16 1247 instead of ax-17 1007.
|- (-. A.x x = y -> (z = y -> A.x z = y))
Theorem	19.23adv 1251	Deduction from Theorem 19.23 of [Margaris] p. 90.
|- (ph -> (ps -> ch))   =>   |- (ph -> (E.xps -> ch))
Theorem	ax11v2 1252	Recovery of ax11o 1254 from ax11v 1303 without using ax-11 1003. The hypothesis is even weaker than ax11v 1303, with z both distinct from x and not occurring in ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1254.
|- (x = z -> (ph -> A.x(x = z -> ph)))   =>   |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Theorem	ax11a2 1253	Derive ax-11o 1255 from a hypothesis in the form of ax-11 1003. The hypothesis is even weaker than ax-11 1003, with z both distinct from x and not occurring in ph. Thus the hypothesis provides an alternate axiom that can be used in place of ax11o 1254. As theorem ax11 1256 shows, the distinct variable conditions are optional. An open problem is whether ax11o 1254 can be derived from ax-11 1003 without relying on ax-17 1007.
|- (x = z -> (A.zph -> A.x(x = z -> ph)))   =>   |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Derive the original axiom of variable substitution ax-11o
Theorem	ax11o 1254	Derivation of set.mm's original ax-11o 1255 from the shorter ax-11 1003 that has replaced it. An open problem is whether this theorem can be proved without relying on ax-16 1247 or ax-17 1007. Another open problem is whether this theorem can be proved without relying on ax-12 1004 (see note in a12study 1417). Theorem ax11 1256 shows the reverse derivation of ax-11 1003 from ax-11o 1255. This theorem should not be referenced in any proof. Instead, use ax-11o 1255 below so that theorems needing ax-11o 1255 can be more easily identified.
|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Axiom	ax-11o 1255	Axiom ax-11o 1255 ("o" for "old") was the original version of ax-11 1003, before it was discovered (in Jan. 2007) that the shorter ax-11 1003 could replace it. It appears as Axiom scheme C15' in [Megill] p. 448 (p. 16 of the preprint). It is based on Lemma 16 of [Tarski] p. 70 and Axiom C8 of [Monk2] p. 105, from which it can be proved by cases. To understand this theorem more easily, think of "-. A.xx = y ->..." as informally meaning "if x and y are distinct variables then..." The antecedent becomes false if the same variable is substituted for x and y, ensuring the theorem is sound whenever this is the case. In some later theorems, we call an antecedent of the form -. A.xx = y a "distinctor." This axiom is redundant, as shown by theorem ax11o 1254.
|- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
Theorem	ax11 1256	Rederivation of axiom ax-11 1003 from the orginal version, ax-11o 1255. See theorem ax11o 1254 for the derivation of ax-11o 1255 from ax-11 1003. This theorem should not be referenced in any proof. Instead, use ax-11 1003 above so that uses of ax-11 1003 can be more easily identified.
|- (x = y -> (A.yph -> A.x(x = y -> ph)))
Theorems without distinct variables that use axiom ax-11o
Theorem	ax11b 1257	A bidirectional version of ax-11o 1255.
|- ((-. A.x x = y /\ x = y) -> (ph <-> A.x(x = y -> ph)))
Theorem	equs5 1258	Lemma used in proofs of substitution properties.
|- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))
Theorem	sb3 1259	One direction of a simplified definition of substitution when variables are distinct.
|- (-. A.x x = y -> (E.x(x = y /\ ph) -> [y / x]ph))
Theorem	sb4 1260	One direction of a simplified definition of substitution when variables are distinct.
|- (-. A.x x = y -> ([y / x]ph -> A.x(x = y -> ph)))
Theorem	sb4b 1261	Simplified definition of substitution when variables are distinct.
|- (-. A.x x = y -> ([y / x]ph <-> A.x(x = y -> ph)))
Theorem	dfsb2 1262	An alternate definition of proper substitution that, like df-sb 1209, mixes free and bound variables to avoid distinct variable requirements.
|- ([y / x]ph <-> ((x = y /\ ph) \/ A.x(x = y -> ph)))
Theorem	dfsb3 1263	An alternate definition of proper substitution df-sb 1209 that uses only primitive connectives (no defined terms) on the right-hand side.
|- ([y / x]ph <-> ((x = y -> -. ph) -> A.x(x = y -> ph)))
Theorem	hbsb2 1264	Bound-variable hypothesis builder for substitution.
|- (-. A.x x = y -> ([y / x]ph -> A.x[y / x]ph))
Theorem	sbequi 1265	An equality theorem for substitution.
|- (x = y -> ([x / z]ph -> [y / z]ph))
Theorem	sbequ 1266	An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint).
|- (x = y -> ([x / z]ph <-> [y / z]ph))
Theorem	drsb2 1267	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint).
|- (A.x x = y -> ([x / z]ph <-> [y / z]ph))
Theorem	sbn 1268	Negation inside and outside of substitution are equivalent.
|- ([y / x] -. ph <-> -. [y / x]ph)
Theorem	sbi1 1269	Removal of implication from substitution.
|- ([y / x](ph -> ps) -> ([y / x]ph -> [y / x]ps))
Theorem	sbi2 1270	Introduction of implication into substitution.
|- (([y / x]ph -> [y / x]ps) -> [y / x](ph -> ps))
Theorem	sbim 1271	Implication inside and outside of substitution are equivalent.
|- ([y / x](ph -> ps) <-> ([y / x]ph -> [y / x]ps))
Theorem	sbor 1272	Logical OR inside and outside of substitution are equivalent.
|- ([y / x](ph \/ ps) <-> ([y / x]ph \/ [y / x]ps))
Theorem	sb19.21 1273	Substitution with a variable not free in antecedent affects only the consequent.
|- (ph -> A.xph)   =>   |- ([y / x](ph -> ps) <-> (ph -> [y / x]ps))
Theorem	sban 1274	Conjunction inside and outside of a substitution are equivalent.
|- ([y / x](ph /\ ps) <-> ([y / x]ph /\ [y / x]ps))
Theorem	sb3an 1275	Conjunction inside and outside of a substitution are equivalent.
|- ([y / x](ph /\ ps /\ ch) <-> ([y / x]ph /\ [y / x]ps /\ [y / x]ch))
Theorem	sbbi 1276	Equivalence inside and outside of a substitution are equivalent.
|- ([y / x](ph <-> ps) <-> ([y / x]ph <-> [y / x]ps))
Theorem	sblbis 1277	Introduce left biconditional inside of a substitution.
|- ([y / x]ph <-> ps)   =>   |- ([y / x](ch <-> ph) <-> ([y / x]ch <-> ps))
Theorem	sbrbis 1278	Introduce right biconditional inside of a substitution.
|- ([y / x]ph <-> ps)   =>   |- ([y / x](ph <-> ch) <-> (ps <-> [y / x]ch))
Theorem	sbrbif 1279	Introduce right biconditional inside of a substitution.
|- (ch -> A.xch)   &   |- ([y / x]ph <-> ps)   =>   |- ([y / x](ph <-> ch) <-> (ps <-> ch))
Theorem	a4sbe 1280	A specialization theorem.
|- ([y / x]ph -> E.xph)
Theorem	a4sbim 1281	Specialization of implication.
|- (A.x(ph -> ps) -> ([y / x]ph -> [y / x]ps))
Theorem	a4sbbi 1282	Specialization of biconditional.
|- (A.x(ph <-> ps) -> ([y / x]ph <-> [y / x]ps))
Theorem	sbbid 1283	Deduction substituting both sides of a biconditional.
|- (ph -> A.xph)   &   |- (ph -> (ps <-> ch))   =>   |- (ph -> ([y / x]ps <-> [y / x]ch))
Theorem	sbequ8 1284	Elimination of equality from antecedent after substitution.
|- ([y / x]ph <-> [y / x](x = y -> ph))
Theorem	sbf3t 1285	Substitution has no effect on a non-free variable.
|- (A.x(ph -> A.xph) -> ([y / x]ph <-> ph))
Theorem	hbsb4 1286	A variable not free remains so after substitution with a distinct variable.
|- (ph -> A.zph)   =>   |- (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph))
Theorem	hbsb4t 1287	A variable not free remains so after substitution with a distinct variable (closed form of hbsb4 1286).
|- (A.xA.z(ph -> A.zph) -> (-. A.z z = y -> ([y / x]ph -> A.z[y / x]ph)))
Theorem	dvelimf 1288	Version of dvelim 1391 without any variable restrictions.
|- (ph -> A.xph)   &   |- (ps -> A.zps)   &   |- (z = y -> (ph <-> ps))   =>   |- (-. A.x x = y -> (ps -> A.xps))
Theorem	dvelimdf 1289	Deduction form of dvelimf 1288. This version may be useful if we want to avoid ax-17 1007 and use ax-16 1247 instead.
|- (ph -> A.xph)   &   |- (ph -> A.zph)   &   |- (ph -> (ps -> A.xps))   &   |- (ph -> (ch -> A.zch))   &   |- (ph -> (z = y -> (ps <-> ch)))   =>   |- (ph -> (-. A.x x = y -> (ch -> A.xch)))
Theorem	sbco 1290	A composition law for substitution.
|- ([y / x][x / y]ph <-> [y / x]ph)
Theorem	sbid2 1291	An identity law for substitution.
|- (ph -> A.xph)   =>   |- ([y / x][x / y]ph <-> ph)
Theorem	sbidm 1292	An idempotent law for substitution.
|- ([y / x][y / x]ph <-> [y / x]ph)
Theorem	sbco2 1293	A composition law for substitution.
|- (ph -> A.zph)   =>   |- ([y / z][z / x]ph <-> [y / x]ph)
Theorem	sbco2d 1294	A composition law for substitution.
|- (ph -> A.xph)   &   |- (ph -> A.zph)   &   |- (ph -> (ps -> A.zps))   =>   |- (ph -> ([y / z][z / x]ps <-> [y / x]ps))
Theorem	sbco3 1295	A composition law for substitution.
|- ([z / y][y / x]ph <-> [z / x][x / y]ph)
Theorem	sbcom 1296	A commutativity law for substitution.
|- ([y / z][y / x]ph <-> [y / x][y / z]ph)
Theorem	sb5rf 1297	Reversed substitution.
|- (ph -> A.yph)   =>   |- (ph <-> E.y(y = x /\ [y / x]ph))
Theorem	sb6rf 1298	Reversed substitution.
|- (ph -> A.yph)   =>   |- (ph <-> A.y(y = x -> [y / x]ph))
Theorem	sb8 1299	Substitution of variable in universal quantifier.
|- (ph -> A.yph)   =>   |- (A.xph <-> A.y[y / x]ph)
Theorem	sb8e 1300	Substitution of variable in existential quantifier.
|- (ph -> A.yph)   =>   |- (E.xph <-> E.y[y / x]ph)