P. 18 of Theorem List - Intuitionistic Logic Explorer

Theorem List for Intuitionistic Logic Explorer - 1701-1800   *Has distinct variable group(s)

Type	Label	Description
Statement
Axiom	ax-11o 1701	Axiom ax-11o 1701 ("o" for "old") was the original version of ax-11 1394, before it was discovered (in Jan. 2007) that the shorter ax-11 1394 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 "¬ ∀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 ¬ ∀xx = y a "distinctor." This axiom is redundant, as shown by theorem ax11o 1700. This axiom is obsolete and should no longer be used. It is proved above as theorem ax11o 1700. (Contributed by NM, 5-Aug-1993.) (New usage is discouraged.)
⊢ (¬ ∀x x = y → (x = y → (φ → ∀x(x = y → φ))))
1.4.3  More theorems related to ax-11 and substitution
Theorem	albidv 1702*	Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∀xψ ↔ ∀xχ))
Theorem	exbidv 1703*	Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 5-Aug-1993.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃xψ ↔ ∃xχ))
Theorem	ax11b 1704	A bidirectional version of ax-11o 1701. (Contributed by NM, 30-Jun-2006.)
⊢ ((¬ ∀x x = y ∧ x = y) → (φ ↔ ∀x(x = y → φ)))
Theorem	ax11v 1705*	This is a version of ax-11o 1701 when the variables are distinct. Axiom (C8) of [Monk2] p. 105. (Contributed by NM, 5-Aug-1993.) (Revised by Jim Kingdon, 15-Dec-2017.)
⊢ (x = y → (φ → ∀x(x = y → φ)))
Theorem	ax11ev 1706*	Analogue to ax11v 1705 for existential quantification. (Contributed by Jim Kingdon, 9-Jan-2018.)
⊢ (x = y → (∃x(x = y ∧ φ) → φ))
Theorem	equs5 1707	Lemma used in proofs of substitution properties. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀x x = y → (∃x(x = y ∧ φ) → ∀x(x = y → φ)))
Theorem	equs5or 1708	Lemma used in proofs of substitution properties. Like equs5 1707 but, in intuitionistic logic, replacing negation and implication with disjunction makes this a stronger result. (Contributed by Jim Kingdon, 2-Feb-2018.)
⊢ (∀x x = y ∨ (∃x(x = y ∧ φ) → ∀x(x = y → φ)))
Theorem	sb3 1709	One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀x x = y → (∃x(x = y ∧ φ) → [y / x]φ))
Theorem	sb4 1710	One direction of a simplified definition of substitution when variables are distinct. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀x x = y → ([y / x]φ → ∀x(x = y → φ)))
Theorem	sb4or 1711	One direction of a simplified definition of substitution when variables are distinct. Similar to sb4 1710 but stronger in intuitionistic logic. (Contributed by Jim Kingdon, 2-Feb-2018.)
⊢ (∀x x = y ∨ ∀x([y / x]φ → ∀x(x = y → φ)))
Theorem	sb4b 1712	Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)
⊢ (¬ ∀x x = y → ([y / x]φ ↔ ∀x(x = y → φ)))
Theorem	sb4bor 1713	Simplified definition of substitution when variables are distinct, expressed via disjunction. (Contributed by Jim Kingdon, 18-Mar-2018.)
⊢ (∀x x = y ∨ ∀x([y / x]φ ↔ ∀x(x = y → φ)))
Theorem	hbsb2 1714	Bound-variable hypothesis builder for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (¬ ∀x x = y → ([y / x]φ → ∀x[y / x]φ))
Theorem	nfsb2or 1715	Bound-variable hypothesis builder for substitution. Similar to hbsb2 1714 but in intuitionistic logic a disjunction is stronger than an implication. (Contributed by Jim Kingdon, 2-Feb-2018.)
⊢ (∀x x = y ∨ Ⅎx[y / x]φ)
Theorem	sbequilem 1716	Propositional logic lemma used in the sbequi 1717 proof. (Contributed by Jim Kingdon, 1-Feb-2018.)
⊢ (φ ∨ (ψ → (χ → θ)))    &   ⊢ (τ ∨ (ψ → (θ → η)))    ⇒   ⊢ (φ ∨ (τ ∨ (ψ → (χ → η))))
Theorem	sbequi 1717	An equality theorem for substitution. (Contributed by NM, 5-Aug-1993.) (Proof modified by Jim Kingdon, 1-Feb-2018.)
⊢ (x = y → ([x / z]φ → [y / z]φ))
Theorem	sbequ 1718	An equality theorem for substitution. Used in proof of Theorem 9.7 in [Megill] p. 449 (p. 16 of the preprint). (Contributed by NM, 5-Aug-1993.)
⊢ (x = y → ([x / z]φ ↔ [y / z]φ))
Theorem	drsb2 1719	Formula-building lemma for use with the Distinctor Reduction Theorem. Part of Theorem 9.4 of [Megill] p. 448 (p. 16 of preprint). (Contributed by NM, 27-Feb-2005.)
⊢ (∀x x = y → ([x / z]φ ↔ [y / z]φ))
Theorem	spsbe 1720	A specialization theorem, mostly the same as Theorem 19.8 of [Margaris] p. 89. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 29-Dec-2017.)
⊢ ([y / x]φ → ∃xφ)
Theorem	spsbim 1721	Specialization of implication. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
⊢ (∀x(φ → ψ) → ([y / x]φ → [y / x]ψ))
Theorem	spsbbi 1722	Specialization of biconditional. (Contributed by NM, 5-Aug-1993.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
⊢ (∀x(φ ↔ ψ) → ([y / x]φ ↔ [y / x]ψ))
Theorem	sbbid 1723	Deduction substituting both sides of a biconditional. (Contributed by NM, 5-Aug-1993.)
⊢ (φ → ∀xφ)    &   ⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → ([y / x]ψ ↔ [y / x]χ))
Theorem	sbequ8 1724	Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) (Proof revised by Jim Kingdon, 20-Jan-2018.)
⊢ ([y / x]φ ↔ [y / x](x = y → φ))
Theorem	sbft 1725	Substitution has no effect on a non-free variable. (Contributed by NM, 30-May-2009.) (Revised by Mario Carneiro, 12-Oct-2016.) (Proof shortened by Wolf Lammen, 3-May-2018.)
⊢ (Ⅎxφ → ([y / x]φ ↔ φ))
Theorem	sbid2h 1726	An identity law for substitution. (Contributed by NM, 5-Aug-1993.)
⊢ (φ → ∀xφ)    ⇒   ⊢ ([y / x][x / y]φ ↔ φ)
Theorem	sbid2 1727	An identity law for substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)
⊢ Ⅎxφ    ⇒   ⊢ ([y / x][x / y]φ ↔ φ)
Theorem	sbidm 1728	An idempotent law for substitution. (Contributed by NM, 30-Jun-1994.) (Proof rewritten by Jim Kingdon, 21-Jan-2018.)
⊢ ([y / x][y / x]φ ↔ [y / x]φ)
Theorem	sb5rf 1729	Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (φ → ∀yφ)    ⇒   ⊢ (φ ↔ ∃y(y = x ∧ [y / x]φ))
Theorem	sb6rf 1730	Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (φ → ∀yφ)    ⇒   ⊢ (φ ↔ ∀y(y = x → [y / x]φ))
Theorem	sb8h 1731	Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
⊢ (φ → ∀yφ)    ⇒   ⊢ (∀xφ ↔ ∀y[y / x]φ)
Theorem	sb8eh 1732	Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Proof rewritten by Jim Kingdon, 15-Jan-2018.)
⊢ (φ → ∀yφ)    ⇒   ⊢ (∃xφ ↔ ∃y[y / x]φ)
Theorem	sb8 1733	Substitution of variable in universal quantifier. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
⊢ Ⅎyφ    ⇒   ⊢ (∀xφ ↔ ∀y[y / x]φ)
Theorem	sb8e 1734	Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Jim Kingdon, 15-Jan-2018.)
⊢ Ⅎyφ    ⇒   ⊢ (∃xφ ↔ ∃y[y / x]φ)
1.4.4  Predicate calculus with distinct variables (cont.)
Theorem	ax16i 1735*	Inference with ax-16 1692 as its conclusion, that doesn't require ax-10 1393, ax-11 1394, or ax-12 1399 for its proof. The hypotheses may be eliminable without one or more of these axioms in special cases. (Contributed by NM, 20-May-2008.)
⊢ (x = z → (φ ↔ ψ))    &   ⊢ (ψ → ∀xψ)    ⇒   ⊢ (∀x x = y → (φ → ∀xφ))
Theorem	ax16ALT 1736*	Version of ax16 1691 that doesn't require ax-10 1393 or ax-12 1399 for its proof. (Contributed by NM, 17-May-2008.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀x x = y → (φ → ∀xφ))
Theorem	spv 1737*	Specialization, using implicit substitition. (Contributed by NM, 30-Aug-1993.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∀xφ → ψ)
Theorem	spimev 1738*	Distinct-variable version of spime 1626. (Contributed by NM, 5-Aug-1993.)
⊢ (x = y → (φ → ψ))    ⇒   ⊢ (φ → ∃xψ)
Theorem	speiv 1739*	Inference from existential specialization, using implicit substitition. (Contributed by NM, 19-Aug-1993.)
⊢ (x = y → (φ ↔ ψ))    &   ⊢ ψ    ⇒   ⊢ ∃xφ
Theorem	equvin 1740*	A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 5-Aug-1993.)
⊢ (x = y ↔ ∃z(x = z ∧ z = y))
Theorem	a16g 1741*	A generalization of axiom ax-16 1692. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (∀x x = y → (φ → ∀zφ))
Theorem	a16gb 1742*	A generalization of axiom ax-16 1692. (Contributed by NM, 5-Aug-1993.)
⊢ (∀x x = y → (φ ↔ ∀zφ))
Theorem	a16nf 1743*	If there is only one element in the universe, then everything satisfies Ⅎ. (Contributed by Mario Carneiro, 7-Oct-2016.)
⊢ (∀x x = y → Ⅎzφ)
Theorem	2albidv 1744*	Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∀x∀yψ ↔ ∀x∀yχ))
Theorem	2exbidv 1745*	Formula-building rule for 2 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃x∃yψ ↔ ∃x∃yχ))
Theorem	3exbidv 1746*	Formula-building rule for 3 existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃x∃y∃zψ ↔ ∃x∃y∃zχ))
Theorem	4exbidv 1747*	Formula-building rule for 4 existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
⊢ (φ → (ψ ↔ χ))    ⇒   ⊢ (φ → (∃x∃y∃z∃wψ ↔ ∃x∃y∃z∃wχ))
Theorem	19.9v 1748*	Special case of Theorem 19.9 of [Margaris] p. 89. (Contributed by NM, 28-May-1995.) (Revised by NM, 21-May-2007.)
⊢ (∃xφ ↔ φ)
Theorem	exlimdd 1749	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 9-Feb-2017.)
⊢ Ⅎxφ    &   ⊢ Ⅎxχ    &   ⊢ (φ → ∃xψ)    &   ⊢ ((φ ∧ ψ) → χ)    ⇒   ⊢ (φ → χ)
Theorem	19.21v 1750*	Special case of Theorem 19.21 of [Margaris] p. 90. Notational convention: We sometimes suffix with "v" the label of a theorem eliminating a hypothesis such as (φ → ∀xφ) in 19.21 1472 via the use of distinct variable conditions combined with ax-17 1416. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a hypothesis to eliminate the need for the distinct variable condition; e.g. euf 1902 derived from df-eu 1900. The "f" stands for "not free in" which is less restrictive than "does not occur in." (Contributed by NM, 5-Aug-1993.)
⊢ (∀x(φ → ψ) ↔ (φ → ∀xψ))
Theorem	alrimiv 1751*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (φ → ψ)    ⇒   ⊢ (φ → ∀xψ)
Theorem	alrimivv 1752*	Inference from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
⊢ (φ → ψ)    ⇒   ⊢ (φ → ∀x∀yψ)
Theorem	alrimdv 1753*	Deduction from Theorem 19.21 of [Margaris] p. 90. (Contributed by NM, 10-Feb-1997.)
⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → (ψ → ∀xχ))
Theorem	nfdv 1754*	Apply the definition of not-free in a context. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (φ → (ψ → ∀xψ))    ⇒   ⊢ (φ → Ⅎxψ)
Theorem	2ax17 1755*	Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
⊢ (φ → ∀x∀yφ)
Theorem	alimdv 1756*	Deduction from Theorem 19.20 of [Margaris] p. 90. (Contributed by NM, 3-Apr-1994.)
⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → (∀xψ → ∀xχ))
Theorem	eximdv 1757*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-1994.)
⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → (∃xψ → ∃xχ))
Theorem	2alimdv 1758*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 27-Apr-2004.)
⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → (∀x∀yψ → ∀x∀yχ))
Theorem	2eximdv 1759*	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 3-Aug-1995.)
⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → (∃x∃yψ → ∃x∃yχ))
Theorem	19.23v 1760*	Special case of Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 28-Jun-1998.)
⊢ (∀x(φ → ψ) ↔ (∃xφ → ψ))
Theorem	19.23vv 1761*	Theorem 19.23 of [Margaris] p. 90 extended to two variables. (Contributed by NM, 10-Aug-2004.)
⊢ (∀x∀y(φ → ψ) ↔ (∃x∃yφ → ψ))
Theorem	sb56 1762*	Two equivalent ways of expressing the proper substitution of y for x in φ, when x and y are distinct. Theorem 6.2 of [Quine] p. 40. The proof does not involve df-sb 1643. (Contributed by NM, 14-Apr-2008.)
⊢ (∃x(x = y ∧ φ) ↔ ∀x(x = y → φ))
Theorem	sb6 1763*	Equivalence for substitution. Compare Theorem 6.2 of [Quine] p. 40. Also proved as Lemmas 16 and 17 of [Tarski] p. 70. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
⊢ ([y / x]φ ↔ ∀x(x = y → φ))
Theorem	sb5 1764*	Equivalence for substitution. Similar to Theorem 6.1 of [Quine] p. 40. (Contributed by NM, 18-Aug-1993.) (Revised by NM, 14-Apr-2008.)
⊢ ([y / x]φ ↔ ∃x(x = y ∧ φ))
Theorem	sbnv 1765*	Version of sbn 1823 where x and y are distinct. (Contributed by Jim Kingdon, 18-Dec-2017.)
⊢ ([y / x] ¬ φ ↔ ¬ [y / x]φ)
Theorem	sbanv 1766*	Version of sban 1826 where x and y are distinct. (Contributed by Jim Kingdon, 24-Dec-2017.)
⊢ ([y / x](φ ∧ ψ) ↔ ([y / x]φ ∧ [y / x]ψ))
Theorem	sborv 1767*	Version of sbor 1825 where x and y are distinct. (Contributed by Jim Kingdon, 3-Feb-2018.)
⊢ ([y / x](φ ∨ ψ) ↔ ([y / x]φ ∨ [y / x]ψ))
Theorem	sbi1v 1768*	Forward direction of sbimv 1770. (Contributed by Jim Kingdon, 25-Dec-2017.)
⊢ ([y / x](φ → ψ) → ([y / x]φ → [y / x]ψ))
Theorem	sbi2v 1769*	Reverse direction of sbimv 1770. (Contributed by Jim Kingdon, 18-Jan-2018.)
⊢ (([y / x]φ → [y / x]ψ) → [y / x](φ → ψ))
Theorem	sbimv 1770*	Intuitionistic proof of sbim 1824 where x and y are distinct. (Contributed by Jim Kingdon, 18-Jan-2018.)
⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → [y / x]ψ))
Theorem	sblimv 1771*	Version of sblim 1828 where x and y are distinct. (Contributed by Jim Kingdon, 19-Jan-2018.)
⊢ (ψ → ∀xψ)    ⇒   ⊢ ([y / x](φ → ψ) ↔ ([y / x]φ → ψ))
Theorem	pm11.53 1772*	Theorem *11.53 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∀x∀y(φ → ψ) ↔ (∃xφ → ∀yψ))
Theorem	exlimivv 1773*	Inference from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 1-Aug-1995.)
⊢ (φ → ψ)    ⇒   ⊢ (∃x∃yφ → ψ)
Theorem	exlimdvv 1774*	Deduction from Theorem 19.23 of [Margaris] p. 90. (Contributed by NM, 31-Jul-1995.)
⊢ (φ → (ψ → χ))    ⇒   ⊢ (φ → (∃x∃yψ → χ))
Theorem	exlimddv 1775*	Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
⊢ (φ → ∃xψ)    &   ⊢ ((φ ∧ ψ) → χ)    ⇒   ⊢ (φ → χ)
Theorem	19.27v 1776*	Theorem 19.27 of [Margaris] p. 90. (Contributed by NM, 3-Jun-2004.)
⊢ (∀x(φ ∧ ψ) ↔ (∀xφ ∧ ψ))
Theorem	19.28v 1777*	Theorem 19.28 of [Margaris] p. 90. (Contributed by NM, 25-Mar-2004.)
⊢ (∀x(φ ∧ ψ) ↔ (φ ∧ ∀xψ))
Theorem	19.36aiv 1778*	Inference from Theorem 19.36 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ ∃x(φ → ψ)    ⇒   ⊢ (∀xφ → ψ)
Theorem	19.41v 1779*	Special case of Theorem 19.41 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (∃x(φ ∧ ψ) ↔ (∃xφ ∧ ψ))
Theorem	19.41vv 1780*	Theorem 19.41 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 30-Apr-1995.)
⊢ (∃x∃y(φ ∧ ψ) ↔ (∃x∃yφ ∧ ψ))
Theorem	19.41vvv 1781*	Theorem 19.41 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 30-Apr-1995.)
⊢ (∃x∃y∃z(φ ∧ ψ) ↔ (∃x∃y∃zφ ∧ ψ))
Theorem	19.41vvvv 1782*	Theorem 19.41 of [Margaris] p. 90 with 4 quantifiers. (Contributed by FL, 14-Jul-2007.)
⊢ (∃w∃x∃y∃z(φ ∧ ψ) ↔ (∃w∃x∃y∃zφ ∧ ψ))
Theorem	19.42v 1783*	Special case of Theorem 19.42 of [Margaris] p. 90. (Contributed by NM, 5-Aug-1993.)
⊢ (∃x(φ ∧ ψ) ↔ (φ ∧ ∃xψ))
Theorem	exdistr 1784*	Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
⊢ (∃x∃y(φ ∧ ψ) ↔ ∃x(φ ∧ ∃yψ))
Theorem	19.42vv 1785*	Theorem 19.42 of [Margaris] p. 90 with 2 quantifiers. (Contributed by NM, 16-Mar-1995.)
⊢ (∃x∃y(φ ∧ ψ) ↔ (φ ∧ ∃x∃yψ))
Theorem	19.42vvv 1786*	Theorem 19.42 of [Margaris] p. 90 with 3 quantifiers. (Contributed by NM, 21-Sep-2011.)
⊢ (∃x∃y∃z(φ ∧ ψ) ↔ (φ ∧ ∃x∃y∃zψ))
Theorem	19.42vvvv 1787*	Theorem 19.42 of [Margaris] p. 90 with 4 quantifiers. (Contributed by Jim Kingdon, 23-Nov-2019.)
⊢ (∃w∃x∃y∃z(φ ∧ ψ) ↔ (φ ∧ ∃w∃x∃y∃zψ))
Theorem	exdistr2 1788*	Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
⊢ (∃x∃y∃z(φ ∧ ψ) ↔ ∃x(φ ∧ ∃y∃zψ))
Theorem	3exdistr 1789*	Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (∃x∃y∃z(φ ∧ ψ ∧ χ) ↔ ∃x(φ ∧ ∃y(ψ ∧ ∃zχ)))
Theorem	4exdistr 1790*	Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
⊢ (∃x∃y∃z∃w((φ ∧ ψ) ∧ (χ ∧ θ)) ↔ ∃x(φ ∧ ∃y(ψ ∧ ∃z(χ ∧ ∃wθ))))
Theorem	cbvalv 1791*	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∀xφ ↔ ∀yψ)
Theorem	cbvexv 1792*	Rule used to change bound variables, using implicit substitition. (Contributed by NM, 5-Aug-1993.)
⊢ (x = y → (φ ↔ ψ))    ⇒   ⊢ (∃xφ ↔ ∃yψ)
Theorem	cbval2 1793*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 22-Dec-2003.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 22-Apr-2018.)
⊢ Ⅎzφ    &   ⊢ Ⅎwφ    &   ⊢ Ⅎxψ    &   ⊢ Ⅎyψ    &   ⊢ ((x = z ∧ y = w) → (φ ↔ ψ))    ⇒   ⊢ (∀x∀yφ ↔ ∀z∀wψ)
Theorem	cbvex2 1794*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 14-Sep-2003.) (Revised by Mario Carneiro, 6-Oct-2016.)
⊢ Ⅎzφ    &   ⊢ Ⅎwφ    &   ⊢ Ⅎxψ    &   ⊢ Ⅎyψ    &   ⊢ ((x = z ∧ y = w) → (φ ↔ ψ))    ⇒   ⊢ (∃x∃yφ ↔ ∃z∃wψ)
Theorem	cbval2v 1795*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 4-Feb-2005.)
⊢ ((x = z ∧ y = w) → (φ ↔ ψ))    ⇒   ⊢ (∀x∀yφ ↔ ∀z∀wψ)
Theorem	cbvex2v 1796*	Rule used to change bound variables, using implicit substitution. (Contributed by NM, 26-Jul-1995.)
⊢ ((x = z ∧ y = w) → (φ ↔ ψ))    ⇒   ⊢ (∃x∃yφ ↔ ∃z∃wψ)
Theorem	cbvald 1797*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1890. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Revised by Wolf Lammen, 13-May-2018.)
⊢ Ⅎyφ    &   ⊢ (φ → Ⅎyψ)    &   ⊢ (φ → (x = y → (ψ ↔ χ)))    ⇒   ⊢ (φ → (∀xψ ↔ ∀yχ))
Theorem	cbvexdh 1798*	Deduction used to change bound variables, using implicit substitition, particularly useful in conjunction with dvelim 1890. (Contributed by NM, 2-Jan-2002.) (Proof rewritten by Jim Kingdon, 30-Dec-2017.)
⊢ (φ → ∀yφ)    &   ⊢ (φ → (ψ → ∀yψ))    &   ⊢ (φ → (x = y → (ψ ↔ χ)))    ⇒   ⊢ (φ → (∃xψ ↔ ∃yχ))
Theorem	cbvexd 1799*	Deduction used to change bound variables, using implicit substitution, particularly useful in conjunction with dvelim 1890. (Contributed by NM, 2-Jan-2002.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
⊢ Ⅎyφ    &   ⊢ (φ → Ⅎyψ)    &   ⊢ (φ → (x = y → (ψ ↔ χ)))    ⇒   ⊢ (φ → (∃xψ ↔ ∃yχ))
Theorem	cbvaldva 1800*	Rule used to change the bound variable in a universal quantifier with implicit substitution. Deduction form. (Contributed by David Moews, 1-May-2017.)
⊢ ((φ ∧ x = y) → (ψ ↔ χ))    ⇒   ⊢ (φ → (∀xψ ↔ ∀yχ))