mmtheorems14 - Metamath Proof Explorer

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Statement List for Metamath Proof Explorer - 1301-1400 - Page 14 of 107

Type	Label	Description
Statement
Theorem	19.12vv 1301	Special case of 19.12 1046 where its converse holds.
⊢ (∃x∀y(φ → ψ) ↔ ∀y∃x(φ → ψ))
Theorem	19.37v 1302	Special case of Theorem 19.37 of [Margaris] p. 90.
⊢ (∃x(φ → ψ) ↔ (φ → ∃xψ))
Theorem	19.37aiv 1303	Inference from Theorem 19.37 of [Margaris] p. 90.
⊢ ∃x(φ → ψ)    ⇒   ⊢ (φ → ∃xψ)
Theorem	19.41v 1304	Special case of Theorem 19.41 of [Margaris] p. 90.
⊢ (∃x(φ ⋀ ψ) ↔ (∃xφ ⋀ ψ))
Theorem	19.41vv 1305	Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers.
⊢ (∃x∃y(φ ⋀ ψ) ↔ (∃x∃yφ ⋀ ψ))
Theorem	19.41vvv 1306	Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers.
⊢ (∃x∃y∃z(φ ⋀ ψ) ↔ (∃x∃y∃zφ ⋀ ψ))
Theorem	19.42v 1307	Special case of Theorem 19.42 of [Margaris] p. 90.
⊢ (∃x(φ ⋀ ψ) ↔ (φ ⋀ ∃xψ))
Theorem	exdistr 1308	Distribution of existential quantifiers.
⊢ (∃x∃y(φ ⋀ ψ) ↔ ∃x(φ ⋀ ∃yψ))
Theorem	19.42vv 1309	Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers.
⊢ (∃x∃y(φ ⋀ ψ) ↔ (φ ⋀ ∃x∃yψ))
Theorem	exdistr2 1310	Distribution of existential quantifiers.
⊢ (∃x∃y∃z(φ ⋀ ψ) ↔ ∃x(φ ⋀ ∃y∃zψ))
Theorem	3exdistr 1311	Distribution of existential quantifiers.
⊢ (∃x∃y∃z(φ ⋀ ψ ⋀ χ) ↔ ∃x(φ ⋀ ∃y(ψ ⋀ ∃zχ)))
Theorem	4exdistr 1312	Distribution of existential quantifiers.
⊢ (∃x∃y∃z∃w((φ ⋀ ψ) ⋀ (χ ⋀ θ)) ↔ ∃x(φ ⋀ ∃y(ψ ⋀ ∃z(χ ⋀ ∃wθ))))
Theorem	cbvalv 1313	Rule used to change bound variables with implicit substitution.
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∀xφ ↔ ∀yψ)
Theorem	cbvexv 1314	Rule used to change bound variables with implicit substitution.
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃xφ ↔ ∃yψ)
Theorem	cbval2 1315	Rule used to change bound variables with implicit substitution.
⊢ (φ → ∀zφ)    &   ⊢ (φ → ∀wφ)    &   ⊢ (ψ → ∀xψ)    &   ⊢ (ψ → ∀yψ)    &   ⊢ ((x = z ⋀ y = w) → (φ ↔ ψ))    ⇒   ⊢ (∀x∀yφ ↔ ∀z∀wψ)
Theorem	cbvex2 1316	Rule used to change bound variables with implicit substitution.
⊢ (φ → ∀zφ)    &   ⊢ (φ → ∀wφ)    &   ⊢ (ψ → ∀xψ)    &   ⊢ (ψ → ∀yψ)    &   ⊢ ((x = z ⋀ y = w) → (φ ↔ ψ))    ⇒   ⊢ (∃x∃yφ ↔ ∃z∃wψ)
Theorem	cbval2v 1317	Rule used to change bound variables with implicit substitution.
⊢ ((x = z ⋀ y = w) → (φ ↔ ψ))    ⇒   ⊢ (∀x∀yφ ↔ ∀z∀wψ)
Theorem	cbvex2v 1318	Rule used to change bound variables with implicit substitution.
⊢ ((x = z ⋀ y = w) → (φ ↔ ψ))    ⇒   ⊢ (∃x∃yφ ↔ ∃z∃wψ)
Theorem	cbvald 1319	Deduction used to change bound variables with implicit substitution, particularly useful in conjunction with dvelim 1351.
⊢ (φ → ∀yφ)    &   ⊢ (φ → (ψ → ∀yψ))    &   ⊢ (φ → (x = y → (ψ ↔ χ)))    ⇒   ⊢ (φ → (∀xψ ↔ ∀yχ))
Theorem	cbvexd 1320	Deduction used to change bound variables with implicit substitution, particularly useful in conjunction with dvelim 1351.
⊢ (φ → ∀yφ)    &   ⊢ (φ → (ψ → ∀yψ))    &   ⊢ (φ → (x = y → (ψ ↔ χ)))    ⇒   ⊢ (φ → (∃xψ ↔ ∃yχ))
Theorem	cbvex4v 1321	Rule used to change bound variables with implicit substitution.
⊢ ((x = v ⋀ y = u) → (φ ↔ ψ))    &   ⊢ ((z = f ⋀ w = g) → (ψ ↔ χ))    ⇒   ⊢ (∃x∃y∃z∃wφ ↔ ∃v∃u∃f∃gχ)
Theorem	eeanv 1322	Rearrange existential quantifiers.
⊢ (∃x∃y(φ ⋀ ψ) ↔ (∃xφ ⋀ ∃yψ))
Theorem	eeeanv 1323	Rearrange existential quantifiers.
⊢ (∃x∃y∃z(φ ⋀ ψ ⋀ χ) ↔ (∃xφ ⋀ ∃yψ ⋀ ∃zχ))
Theorem	ee4anv 1324	Rearrange existential quantifiers.
⊢ (∃x∃y∃z∃w(φ ⋀ ψ) ↔ (∃x∃yφ ⋀ ∃z∃wψ))
Theorem	nexdv 1325	Deduction for generalization rule for negated wff.
⊢ (φ → ¬ ψ)    ⇒   ⊢ (φ → ¬ ∃xψ)
Theorem	chvarv 1326	Implicit substitution of y for x into a theorem.
⊢ (x = y → (φ ↔ ψ))    &   ⊢ φ    ⇒   ⊢ ψ
Theorem	cleljust 1327	When the class variables in definition df-clel 1471 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 958 with the class variables in wcel 957.
⊢ (x ∈ y ↔ ∃z(z = x ⋀ z ∈ y))
More substitution theorems
Theorem	equsb3lem 1328	Lemma for equsb3 1329.
Theorem	equsb3 1329	Substitution applied to an atomic wff. (Contributed by Raph Levien and FL, 4-Dec-2005.)
⊢ ([x / y]y = z ↔ x = z)
Theorem	elsb3 1330	Substitution applied to an atomic membership wff.
⊢ ([x / y]y ∈ z ↔ x ∈ z)
Theorem	hbs1 1331	x is not free in [y / x]φ when x and y are distinct.
⊢ ([y / x]φ → ∀x[y / x]φ)
Theorem	hbsb 1332	If z is not free in φ, it is not free in [y / x]φ when y and z are distinct.
⊢ (φ → ∀zφ)    ⇒   ⊢ ([y / x]φ → ∀z[y / x]φ)
Theorem	sbcom2 1333	Commutativity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint).
⊢ ([w / z][y / x]φ ↔ [y / x][w / z]φ)
Theorem	2sb5 1334	Equivalence for double substitution.
⊢ ([z / x][w / y]φ ↔ ∃x∃y((x = z ⋀ y = w) ⋀ φ))
Theorem	2sb6 1335	Equivalence for double substitution.
⊢ ([z / x][w / y]φ ↔ ∀x∀y((x = z ⋀ y = w) → φ))
Theorem	sb6a 1336	Equivalence for substitution.
⊢ ([y / x]φ ↔ ∀x(x = y → [x / y]φ))
Theorem	2sb5rf 1337	Reversed double substitution.
⊢ (φ → ∀zφ)    &   ⊢ (φ → ∀wφ)    ⇒   ⊢ (φ ↔ ∃z∃w((z = x ⋀ w = y) ⋀ [z / x][w / y]φ))
Theorem	2sb6rf 1338	Reversed double substitution.
⊢ (φ → ∀zφ)    &   ⊢ (φ → ∀wφ)    ⇒   ⊢ (φ ↔ ∀z∀w((z = x ⋀ w = y) → [z / x][w / y]φ))
Theorem	dfsb7 1339	An alternate definition of proper substitution df-sb 1171. By introducing a dummy variable z in the definiens, we are able to eliminate any distinct variable restrictions among the variables x, y, and φ of the definiendum. No distinct variable conflicts arise because z effectively insulates x from y. To achieve this, we use a chain of two substitutions in the form of sb5 1267, first z for x then y for z. Compare Definition 2.1'' of [Quine] p. 17, which is obtained from this theorem by applying df-clab 1463. Theorem sb7f 1340 provides a version where φ and z don't have to be distinct.
⊢ ([y / x]φ ↔ ∃z(z = y ⋀ ∃x(x = z ⋀ φ)))
Theorem	sb7f 1340	This version of dfsb7 1339 does not require that φ and z be distinct. This permits it to be used as a definition for substitution in a formalization that omits the logically redundant axiom ax-17 970 i.e. that doesn't have the concept of a variable not occurring in a wff. (df-sb 1171 is also suitable, but its mixing of free and bound variables is distasteful to some logicians.)
⊢ (φ → ∀zφ)    ⇒   ⊢ ([y / x]φ ↔ ∃z(z = y ⋀ ∃x(x = z ⋀ φ)))
Theorem	sb10f 1341	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.
⊢ (φ → ∀xφ)    ⇒   ⊢ ([y / z]φ ↔ ∃x(x = y ⋀ [x / z]φ))
Theorem	sbid2v 1342	An identity law for substitution. Used in proof of Theorem 9.7 of [Megill] p. 449 (p. 16 of the preprint).
⊢ ([y / x][x / y]φ ↔ φ)
Theorem	sbelx 1343	Elimination of substitution.
⊢ (φ ↔ ∃x(x = y ⋀ [x / y]φ))
Theorem	sbel2x 1344	Elimination of double substitution.
⊢ (φ ↔ ∃x∃y((x = z ⋀ y = w) ⋀ [y / w][x / z]φ))
Theorem	sbal1 1345	A theorem used in elimination of disjoint variable restriction on x and y by replacing it with a distinctor ¬ ∀xx = z.
⊢ (¬ ∀x x = z → ([z / y]∀xφ ↔ ∀x[z / y]φ))
Theorem	sbal 1346	Move universal quantifier in and out of substitution.
⊢ ([z / y]∀xφ ↔ ∀x[z / y]φ)
Theorem	sbex 1347	Move existential quantifier in and out of substitution.
⊢ ([z / y]∃xφ ↔ ∃x[z / y]φ)
Theorem	sbalv 1348	Quantify with new variable inside substitution.
⊢ ([y / x]φ ↔ ψ)    ⇒   ⊢ ([y / x]∀zφ ↔ ∀zψ)
Theorem	exsb 1349	An equivalent expression for existence.
⊢ (∃xφ ↔ ∃y∀x(x = y → φ))
Theorem	2exsb 1350	An equivalent expression for double existence.
⊢ (∃x∃yφ ↔ ∃z∃w∀x∀y((x = z ⋀ y = w) → φ))
Theorem	dvelim 1351	This theorem can be used to eliminate a distinct variable restriction on x and z and replace it with the "distinctor" ¬ ∀xx = y as an antecedent. φ normally has z free and can be read φ(z), and ψ substitutes y for z and can be read φ(y). We don't require that x and y be distinct: if they aren't, the distinctor will become false (in multiple-element domains of discourse) and "protect" the consequent. To obtain a closed-theorem form of this inference, prefix the hypotheses with ∀x∀z, conjoin them, and apply dvelimdf 1250.
⊢ (φ → ∀xφ)    &   ⊢ (z = y → (φ ↔ ψ))    ⇒   ⊢ (¬ ∀x x = y → (ψ → ∀xψ))
Theorem	dvelimALT 1352	Version of dvelim 1351 that doesn't use ax-10 965. (See dvelimfALT 1152 for a version that doesn't use ax-11 966.)
⊢ (φ → ∀xφ)    &   ⊢ (z = y → (φ ↔ ψ))    ⇒   ⊢ (¬ ∀x x = y → (ψ → ∀xψ))
Theorem	dveeq1 1353	Quantifier introduction when one pair of variables is distinct.
⊢ (¬ ∀x x = y → (y = z → ∀x y = z))
Theorem	dveeq1ALT 1354	Version of dveeq1 1353 using ax-16 1209 instead of ax-17 970.
⊢ (¬ ∀x x = y → (y = z → ∀x y = z))
Theorem	dveel1 1355	Quantifier introduction when one pair of variables is distinct.
⊢ (¬ ∀x x = y → (y ∈ z → ∀x y ∈ z))
Theorem	dveel2 1356	Quantifier introduction when one pair of variables is distinct.
⊢ (¬ ∀x x = y → (z ∈ y → ∀x z ∈ y))
Theorem	sbal2 1357	Move quantifier in and out of substitution.
⊢ (¬ ∀x x = y → ([z / y]∀xφ ↔ ∀x[z / y]φ))
Theorem	ax15 1358	Axiom ax-15 1359 is redundant if we assume ax-17 970. Remark 9.6 in [Megill] p. 448 (p. 16 of the preprint), regarding axiom scheme C14'. Note that w 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 1356 and ax-17 970. By the end of the proof it has vanished, and the final theorem has no distinct variable requirements. This theorem should not be referenced in any proof. Instead, use ax-15 1359 below so that theorems needing ax-15 1359 can be more easily identified.
⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x ∈ y → ∀z x ∈ y)))
Axiom	ax-15 1359	Axiom of Quantifier Introduction. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. Axiom scheme C14' in [Megill] p. 448 (p. 16 of the preprint). It is redundant if we include ax-17 970; see theorem ax15 1358. Alternately, ax-17 970 becomes unnecessary in principle with this axiom, but we lose the more powerful metalogic afforded by ax-17 970. We retain ax-15 1359 here to provide completeness for systems with the simpler metalogic that results from omitting ax-17 970, which might be easier to study for some theoretical purposes.
⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x ∈ y → ∀z x ∈ y)))
Theorem	ax17el 1360	Theorem to add distinct quantifier to atomic formula. This theorem demonstrates the induction basis for ax-17 970 considered as a metatheorem.)
⊢ (x ∈ y → ∀z x ∈ y)
Theorem	dveel2ALT 1361	Version of dveel2 1356 using ax-16 1209 instead of ax-17 970.
⊢ (¬ ∀x x = y → (z ∈ y → ∀x z ∈ y))
Theorem	ax11eq 1362	Basis step for constructing a substitution instance of ax-11o 1217 without using ax-11o 1217. Atomic formula for equality predicate.
⊢ (¬ ∀x x = y → (x = y → (z = w → ∀x(x = y → z = w))))
Theorem	ax11el 1363	Basis step for constructing a substitution instance of ax-11o 1217 without using ax-11o 1217. Atomic formula for membership predicate.
⊢ (¬ ∀x x = y → (x = y → (z ∈ w → ∀x(x = y → z ∈ w))))
Theorem	ax11f 1364	Basis step for constructing a substitution instance of ax-11o 1217 without using ax-11o 1217. We can start with any formula φ in which x is not free.
⊢ (φ → ∀xφ)    ⇒   ⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
Theorem	ax11indn 1365	Induction step for constructing a substitution instance of ax-11o 1217 without using ax-11o 1217. Negation case.
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))    ⇒   ⊢ (¬ ∀x x = y → (x = y → (¬ φ → ∀x(x = y → ¬ φ))))
Theorem	ax11indi 1366	Induction step for constructing a substitution instance of ax-11o 1217 without using ax-11o 1217. Implication case.
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))    &   ⊢ (¬ ∀x x = y → (x = y → (ψ → ∀x(x = y → ψ))))    ⇒   ⊢ (¬ ∀x x = y → (x = y → ((φ → ψ) → ∀x(x = y → (φ → ψ)))))
Theorem	ax11indalem 1367	Lemma for ax11inda2 1369 and ax11inda 1370.
Theorem	ax11inda2ALT 1368	A proof of ax11inda2 1369 that is slightly more direct.
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))    ⇒   ⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))
Theorem	ax11inda2 1369	Induction step for constructing a substitution instance of ax-11o 1217 without using ax-11o 1217. Quantification case. When z and y are distinct, this theorem avoids the dummy variables needed by the more general ax11inda 1370.
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))    ⇒   ⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))
Theorem	ax11inda 1370	Induction step for constructing a substitution instance of ax-11o 1217 without using ax-11o 1217. Quantification case. (When z and y are distinct, ax11inda2 1369 may be used instead to avoid the dummy variable w in the proof.)
⊢ (¬ ∀x x = w → (x = w → (φ → ∀x(x = w → φ))))    ⇒   ⊢ (¬ ∀x x = y → (x = y → (∀zφ → ∀x(x = y → ∀zφ))))
Theorem	a12stdy1 1371	Part of a study related to ax-12 967. The consequent introduces a new variable z. There are no distinct variable restrictions.
⊢ (∀x x = y → ∃x y = z)
Theorem	a12stdy2 1372	Part of a study related to ax-12 967. The consequent is quantified with a different variable. There are no distinct variable restrictions.
⊢ (∀z(z = x ⋀ x = y) → ∀y y = x)
Theorem	a12stdy3 1373	Part of a study related to ax-12 967. The consequent introduces two new variables. There are no distinct variable restrictions.
⊢ (∀z(z = x ⋀ x = y) → ∀v∃y x = w)
Theorem	a12stdy4 1374	Part of a study related to ax-12 967. The second antecedent of ax-12 967 is replaced. There are no distinct variable restrictions.
⊢ (¬ ∀z z = x → (∀y z = x → (x = y → ∀z x = y)))
Theorem	a12lem1 1375	Proof of first hypothesis of a12study 1377.
⊢ (¬ ∀z z = y → (∀z(z = x → z = y) → x = y))
Theorem	a12lem2 1376	Proof of second hypothesis of a12study 1377.
⊢ (∀z(z = x → ¬ z = y) → ¬ x = y)
Theorem	a12study 1377	Rederivation of axiom ax-12 967 from two shorter formulas, without using ax-12 967. See a12lem1 1375 and a12lem2 1376 for the proofs of the hypotheses (using ax-12 967). This is the only known breakdown of ax-12 967 into shorter formulas. See a12studyALT 1378 for an alternate proof. Note that the proof depends on ax-11o 1217, whose proof ax11o 1216 depends on ax-12 967, meaning that we would have to replace ax-11 966 with ax-11o 1217 in an axiomatization that uses the hypotheses in place of ax-12 967. Whether this can be avoided is an open problem.
⊢ (¬ ∀z z = y → (∀z(z = x → z = y) → x = y))    &   ⊢ (∀z(z = x → ¬ z = y) → ¬ x = y)    ⇒   ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
Theorem	a12studyALT 1378	Alternate proof of a12study 1377, also without using ax-12 967.
⊢ (¬ ∀z z = y → (∀z(z = x → z = y) → x = y))    &   ⊢ (∀z(z = x → ¬ z = y) → ¬ x = y)    ⇒   ⊢ (¬ ∀z z = x → (¬ ∀z z = y → (x = y → ∀z x = y)))
Existential uniqueness
Syntax	weu 1379	Extend wff definition to include existential uniqueness ("there exists a unique x such that φ").
wff ∃!xφ
Syntax	wmo 1380	Extend wff definition to include uniqueness ("there exists at most one x such that φ").
wff ∃*xφ
Definition	df-eu 1381	Define existential uniqueness, i.e. "there exists exactly one x such that φ." Definition 10.1 of [BellMachover] p. 97; also Definition *14.02 of [WhiteheadRussell] p. 175. Other possible definitions are given by eu1 1391, eu2 1395, eu3 1396, and eu5 1408 (which in some cases we show with a hypothesis φ → ∀yφ in place of a distinct variable condition on y and φ). Double uniqueness is tricky: ∃!x∃!yφ does not mean "exactly one x and one y" (see 2eu4 1451).
⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
Definition	df-mo 1382	Define "there exists at most one x such that φ." Here we define it in terms of existential uniqueness. Notation of [BellMachover] p. 460, whose definition we show as mo3 1400. For other possible definitions see mo2 1399 and mo4 1402.
⊢ (∃*xφ ↔ (∃xφ → ∃!xφ))
Theorem	euf 1383	A version of the existential uniqueness definition with a hypothesis instead of a distinct variable condition.
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃!xφ ↔ ∃y∀x(φ ↔ x = y))
Theorem	eubid 1384	Formula-building rule for uniqueness quantifier (deduction rule).
⊢ (φ → ∀xφ)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃!xψ ↔ ∃!xχ))
Theorem	eubidv 1385	Formula-building rule for uniqueness quantifier (deduction rule).
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃!xψ ↔ ∃!xχ))
Theorem	eubii 1386	Introduce uniqueness quantifier to both sides of an equivalence.
⊢ (φ ↔ ψ)    ⇒   ⊢ (∃!xφ ↔ ∃!xψ)
Theorem	hbeu1 1387	Bound-variable hypothesis builder for uniqueness.
⊢ (∃!xφ → ∀x∃!xφ)
Theorem	hbeu 1388	Bound-variable hypothesis builder for "at most one." Note that x and y needn't be distinct (this makes the proof more difficult).
⊢ (φ → ∀xφ)    ⇒   ⊢ (∃!yφ → ∀x∃!yφ)
Theorem	sb8eu 1389	Variable substitution in uniqueness quantifier. (This theorem can also be proved without requiring that x and y be distinct, but the proof would be longer.)
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃!xφ ↔ ∃!y[y / x]φ)
Theorem	cbveu 1390	Rule used to change bound variables with implicit substitution.
⊢ (φ → ∀yφ)    &   ⊢ (ψ → ∀xψ)    &   ⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃!xφ ↔ ∃!yψ)
Theorem	eu1 1391	An alternate way to express uniqueness used by some authors. Exercise 2(b) of [Margaris] p. 110.
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃!xφ ↔ ∃x(φ ⋀ ∀y([y / x]φ → x = y)))
Theorem	mo 1392	Equivalent definitions of "there exists at most one."
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃y∀x(φ → x = y) ↔ ∀x∀y((φ ⋀ [y / x]φ) → x = y))
Theorem	euex 1393	Existential uniqueness implies existence.
⊢ (∃!xφ → ∃xφ)
Theorem	eumo0 1394	Existential uniqueness implies "at most one."
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃!xφ → ∃y∀x(φ → x = y))
Theorem	eu2 1395	An alternate way of defining existential uniqueness. Definition 6.10 of [TakeutiZaring] p. 26.
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃!xφ ↔ (∃xφ ⋀ ∀x∀y((φ ⋀ [y / x]φ) → x = y)))
Theorem	eu3 1396	An alternate way to express existential uniqueness.
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃!xφ ↔ (∃xφ ⋀ ∃y∀x(φ → x = y)))
Theorem	euor 1397	Introduce a disjunct into a uniqueness quantifier.
⊢ (φ → ∀xφ)    ⇒   ⊢ ((¬ φ ⋀ ∃!xψ) → ∃!x(φ ⋁ ψ))
Theorem	euorv 1398	Introduce a disjunct into a uniqueness quantifier.
⊢ ((¬ φ ⋀ ∃!xψ) → ∃!x(φ ⋁ ψ))
Theorem	mo2 1399	Alternate definition of "at most one."
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃*xφ ↔ ∃y∀x(φ → x = y))
Theorem	mo3 1400	Alternate definition of "at most one." Definition of [BellMachover] p. 460, except that definition has the side condition that y not occur in φ in place of our hypothesis.
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃*xφ ↔ ∀x∀y((φ ⋀ [y / x]φ) → x = y))