P. 18 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 1701-1800   *Has distinct variable group(s)

Type	Label	Description
Statement
Definition	df-nf 1701	Define the not-free predicate for wffs. This is read "𝑥 is not free in 𝜑". Not-free means that the value of 𝑥 cannot affect the value of 𝜑, e.g., any occurrence of 𝑥 in 𝜑 is effectively bound by a "for all" or something that expands to one (such as "there exists"). In particular, substitution for a variable not free in a wff does not affect its value (sbf 2368). An example of where this is used is stdpc5 2063. See nf5 2102 for an alternate definition which involves nested quantifiers on the same variable. Not-free is a commonly used constraint, so it is useful to have a notation for it. Surprisingly, there is no common formal notation for it, so here we devise one. Our definition lets us work with the not-free notion within the logic itself rather than as a metalogical side condition. To be precise, our definition really means "effectively not free," because it is slightly less restrictive than the usual textbook definition for not-free (which only considers syntactic freedom). For example, 𝑥 is effectively not free in the bare expression 𝑥 = 𝑥 (see nfequid 1927), even though 𝑥 would be considered free in the usual textbook definition, because the value of 𝑥 in the expression 𝑥 = 𝑥 cannot affect the truth of the expression (and thus substitution will not change the result). This definition of not-free tightly ties to the quantifier ∀𝑥. At this state (no axioms restricting quantifiers yet) 'non-free' appears quite arbitrary. Its intended semantics expresses single-valuedness (constness) across a parameter, but is only evolved as much as later axioms assign properties to quantifiers. It seems the definition here is best suited in situations, where axioms are only partially in effect. In particular, this definition more easily carries over to other logic models with weaker axiomization. This predicate only applies to wffs. See df-nfc 2740 for a not-free predicate for class variables. (Contributed by Mario Carneiro, 24-Sep-2016.) Converted to definition. (Revised by BJ, 6-May-2019.)
⊢ (Ⅎ𝑥𝜑 ↔ (∃𝑥𝜑 → ∀𝑥𝜑))
Theorem	nf2 1702	Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ¬ ∃𝑥𝜑))
Theorem	nf3 1703	Alternate definition of non-freeness. (Contributed by BJ, 16-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ (∀𝑥𝜑 ∨ ∀𝑥 ¬ 𝜑))
Theorem	nf4 1704	Alternate definition of non-freeness. This definition uses only primitive symbols. (Contributed by BJ, 16-Sep-2021.)
⊢ (Ⅎ𝑥𝜑 ↔ (¬ ∀𝑥𝜑 → ∀𝑥 ¬ 𝜑))
Theorem	nfi 1705	Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Wolf Lammen, 15-Sep-2021.)
⊢ (∃𝑥𝜑 → ∀𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfri 1706	Consequence of the definition of not-free. (Contributed by Wolf Lammen, 16-Sep-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ (∃𝑥𝜑 → ∀𝑥𝜑)
Theorem	nfd 1707	Deduce that 𝑥 is not free in 𝜓 in a context. (Contributed by Wolf Lammen, 16-Sep-2021.)
⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))    ⇒   ⊢ (𝜑 → Ⅎ𝑥𝜓)
Theorem	nfrd 1708	Consequence of the definition of not-free in a context. (Contributed by Wolf Lammen, 15-Oct-2021.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∀𝑥𝜓))
Theorem	nftht0 1709	Closed form of nfth 1718. (Contributed by Wolf Lammen, 19-Aug-2018.) (Proof shortened by BJ, 16-Sep-2021.)
⊢ (∀𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	nfntht 1710	Closed form of nfnth 1719. (Contributed by BJ, 16-Sep-2021.)
⊢ (¬ ∃𝑥𝜑 → Ⅎ𝑥𝜑)
Theorem	nfntht2 1711	Closed form of nfnth 1719. (Contributed by BJ, 16-Sep-2021.)
⊢ (∀𝑥 ¬ 𝜑 → Ⅎ𝑥𝜑)
Definition	df-nfOLD 1712	Obsolete definition replaced by nf5 2102 as of 3-Oct-2021. This definition is less suitable than df-nf 1701 when ax-10 2006 and ax-12 2034 are not in effect. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
⊢ (Ⅎ𝑥𝜑 ↔ ∀𝑥(𝜑 → ∀𝑥𝜑))
1.4.2  Rule scheme ax-gen (Generalization)
Axiom	ax-gen 1713	Rule of Generalization. The postulated inference rule of predicate calculus. See e.g. Rule 2 of [Hamilton] p. 74. This rule says that if something is unconditionally true, then it is true for all values of a variable. For example, if we have proved 𝑥 = 𝑥, we can conclude ∀𝑥𝑥 = 𝑥 or even ∀𝑦𝑥 = 𝑥. Theorem allt 31570 shows the special case ∀𝑥⊤. Theorem spi 2042 shows we can go the other way also: in other words we can add or remove universal quantifiers from the beginning of any theorem as required. (Contributed by NM, 3-Jan-1993.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥𝜑
Theorem	gen2 1714	Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
⊢ 𝜑    ⇒   ⊢ ∀𝑥∀𝑦𝜑
Theorem	mpg 1715	Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
⊢ (∀𝑥𝜑 → 𝜓)    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	mpgbi 1716	Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
⊢ (∀𝑥𝜑 ↔ 𝜓)    &   ⊢ 𝜑    ⇒   ⊢ 𝜓
Theorem	mpgbir 1717	Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
⊢ (𝜑 ↔ ∀𝑥𝜓)    &   ⊢ 𝜓    ⇒   ⊢ 𝜑
Theorem	nfth 1718	No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
⊢ 𝜑    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfnth 1719	No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
⊢ ¬ 𝜑    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	hbth 1720	No variable is (effectively) free in a theorem. This and later "hypothesis-building" lemmas, with labels starting "hb...", allow us to construct proofs of formulas of the form ⊢ (𝜑 → ∀𝑥𝜑) from smaller formulas of this form. These are useful for constructing hypotheses that state "𝑥 is (effectively) not free in 𝜑." (Contributed by NM, 11-May-1993.)
⊢ 𝜑    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	nftru 1721	The true constant has no free variables. (This can also be proven in one step with nfv 1830, but this proof does not use ax-5 1827.) (Contributed by Mario Carneiro, 6-Oct-2016.)
⊢ Ⅎ𝑥⊤
Theorem	nex 1722	Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
⊢ ¬ 𝜑    ⇒   ⊢ ¬ ∃𝑥𝜑
Theorem	nffal 1723	The false constant has no free variables (see nftru 1721). (Contributed by BJ, 6-May-2019.)
⊢ Ⅎ𝑥⊥
Theorem	sptruw 1724	Version of sp 2041 when 𝜑 is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.)
⊢ 𝜑    ⇒   ⊢ (∀𝑥𝜑 → 𝜑)
Theorem	nfiOLD 1725	Obsolete proof of nf5i 2011 as of 5-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfthOLD 1726	Obsolete proof of nfth 1718 as of 5-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ 𝜑    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfnthOLD 1727	Obsolete proof of nfnth 1719 as of 6-Oct-2021. (Contributed by Mario Carneiro, 6-Dec-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ¬ 𝜑    ⇒   ⊢ Ⅎ𝑥𝜑
1.4.3  Axiom scheme ax-4 (Quantified Implication)
Axiom	ax-4 1728	Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1729 for labeling consistency. It should be used only by alim 1729. (Contributed by NM, 21-May-2008.) Use alim 1729 instead. (New usage is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	alim 1729	Restatement of Axiom ax-4 1728, for labeling consistency. It should be the only theorem using ax-4 1728. (Contributed by NM, 10-Jan-1993.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
Theorem	alimi 1730	Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∀𝑥𝜓)
Theorem	2alimi 1731	Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓)
Theorem	al2im 1732	Closed form of al2imi 1733. Version of alim 1729 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
Theorem	al2imi 1733	Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	alanimi 1734	Variant of al2imi 1733 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
⊢ ((𝜑 ∧ 𝜓) → 𝜒)    ⇒   ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)
Theorem	alimdh 1735	Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1729. (Contributed by NM, 4-Jan-2002.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
Theorem	albi 1736	Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
Theorem	albii 1737	Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓)
Theorem	2albii 1738	Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∀𝑥∀𝑦𝜑 ↔ ∀𝑥∀𝑦𝜓)
Theorem	sylgt 1739	Closed form of sylg 1740. (Contributed by BJ, 2-May-2019.)
⊢ (∀𝑥(𝜓 → 𝜒) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))
Theorem	sylg 1740	A syllogism combined with generalization. Inference associated with sylgt 1739. General form of alrimih 1741. (Contributed by BJ, 4-Oct-2019.)
⊢ (𝜑 → ∀𝑥𝜓)    &   ⊢ (𝜓 → 𝜒)    ⇒   ⊢ (𝜑 → ∀𝑥𝜒)
Theorem	alrimih 1741	Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2062 and 19.21h 2107. Instance of sylg 1740. (Contributed by NM, 9-Jan-1993.) (Revised by BJ, 31-Mar-2021.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜓)
Theorem	hbxfrbi 1742	A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2717 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜓 → ∀𝑥𝜓)    ⇒   ⊢ (𝜑 → ∀𝑥𝜑)
Theorem	alex 1743	Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
Theorem	exnal 1744	Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ (∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
Theorem	2nalexn 1745	Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑)
Theorem	2exnaln 1746	Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑)
Theorem	2nexaln 1747	Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑)
Theorem	alimex 1748	A utility theorem. An interesting case is when the same formula is substituted for both 𝜑 and 𝜓, since then both implications express a type of non-freeness. See also eximal 1698. (Contributed by BJ, 12-May-2019.)
⊢ ((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))
Theorem	aleximi 1749	A variant of al2imi 1733: instead of applying ∀𝑥 quantifiers to the final implication, replace them with ∃𝑥. A shorter proof is possible using nfa1 2015, sps 2043 and eximd 2072, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	alexbii 1750	Biconditional form of aleximi 1749. (Contributed by BJ, 16-Nov-2020.)
⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	exim 1751	Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
Theorem	eximi 1752	Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥𝜑 → ∃𝑥𝜓)
Theorem	2eximi 1753	Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
⊢ (𝜑 → 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓)
Theorem	eximii 1754	Inference associated with eximi 1752. (Contributed by BJ, 3-Feb-2018.)
⊢ ∃𝑥𝜑    &   ⊢ (𝜑 → 𝜓)    ⇒   ⊢ ∃𝑥𝜓
Theorem	ala1 1755	Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)
⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜑))
Theorem	exa1 1756	Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.)
⊢ (∃𝑥𝜑 → ∃𝑥(𝜓 → 𝜑))
Theorem	19.38 1757	Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2061. (Revised by Wolf Lammen, 2-Jan-2018.)
⊢ ((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑 → 𝜓))
Theorem	imnang 1758	Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓))
Theorem	alinexa 1759	A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓))
Theorem	alexn 1760	A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑)
Theorem	2exnexn 1761	Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.)
⊢ (∃𝑥∀𝑦𝜑 ↔ ¬ ∀𝑥∃𝑦 ¬ 𝜑)
Theorem	exbi 1762	Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
Theorem	exbiOLD 1763	Obsolete proof of exbi 1762 as of 16-Nov-2020. (Contributed by NM, 12-Mar-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
Theorem	exbii 1764	Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥𝜑 ↔ ∃𝑥𝜓)
Theorem	2exbii 1765	Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜓)
Theorem	3exbii 1766	Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (∃𝑥∃𝑦∃𝑧𝜑 ↔ ∃𝑥∃𝑦∃𝑧𝜓)
Theorem	nfnt 1767	If 𝑥 is not free in 𝜑, then it is not free in ¬ 𝜑. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) df-nf 1701 changed. (Revised by Wolf Lammen, 4-Oct-2021.)
⊢ (Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
Theorem	nfn 1768	Inference associated with nfnt 1767. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
⊢ Ⅎ𝑥𝜑    ⇒   ⊢ Ⅎ𝑥 ¬ 𝜑
Theorem	nfnd 1769	Deduction associated with nfnt 1767. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜑 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓)
Theorem	nfbii 1770	Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1701 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
⊢ (𝜑 ↔ 𝜓)    ⇒   ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
Theorem	nfxfr 1771	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ Ⅎ𝑥𝜓    ⇒   ⊢ Ⅎ𝑥𝜑
Theorem	nfxfrd 1772	A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
⊢ (𝜑 ↔ 𝜓)    &   ⊢ (𝜒 → Ⅎ𝑥𝜓)    ⇒   ⊢ (𝜒 → Ⅎ𝑥𝜑)
Theorem	exanali 1773	A transformation of quantifiers and logical connectives. (Contributed by NM, 25-Mar-1996.) (Proof shortened by Wolf Lammen, 4-Sep-2014.)
⊢ (∃𝑥(𝜑 ∧ ¬ 𝜓) ↔ ¬ ∀𝑥(𝜑 → 𝜓))
Theorem	exancom 1774	Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑))
Theorem	exan 1775	Place a conjunct in the scope of an existential quantifier. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof shortened by Wolf Lammen, 8-Oct-2021.)
⊢ (∃𝑥𝜑 ∧ 𝜓)    ⇒   ⊢ ∃𝑥(𝜑 ∧ 𝜓)
Theorem	exanOLD 1776	Obsolete proof of exan 1775 as of 8-Oct-2021. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) Reduce axiom dependencies. (Revised by BJ, 7-Jul-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥𝜑 ∧ 𝜓)    ⇒   ⊢ ∃𝑥(𝜑 ∧ 𝜓)
Theorem	alrimdh 1777	Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2062 and 19.21h 2107. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜓 → ∀𝑥𝜓)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒))
Theorem	eximdh 1778	Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 → 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
Theorem	nexdh 1779	Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → ¬ 𝜓)    ⇒   ⊢ (𝜑 → ¬ ∃𝑥𝜓)
Theorem	albidh 1780	Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
Theorem	exbidh 1781	Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	exbidhOLD 1782	Obsolete proof of exbidh 1781 as of 16-Nov-2020. (Contributed by NM, 26-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (𝜑 → ∀𝑥𝜑)    &   ⊢ (𝜑 → (𝜓 ↔ 𝜒))    ⇒   ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
Theorem	exsimpl 1783	Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜑)
Theorem	exsimpr 1784	Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) → ∃𝑥𝜓)
Theorem	19.40 1785	Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
Theorem	19.26 1786	Theorem 19.26 of [Margaris] p. 90. Also Theorem *10.22 of [WhiteheadRussell] p. 147. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
Theorem	19.26-2 1787	Theorem 19.26 1786 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓))
Theorem	19.26-3an 1788	Theorem 19.26 1786 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))
Theorem	19.29 1789	Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1790. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
⊢ ((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
Theorem	19.29r 1790	Variation of 19.29 1789. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
Theorem	19.29rOLD 1791	Obsolete proof of 19.29r 1790 as 12-Nov-2020. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓))
Theorem	19.29r2 1792	Variation of 19.29r 1790 with double quantification. (Contributed by NM, 3-Feb-2005.)
⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))
Theorem	19.29x 1793	Variation of 19.29 1789 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓))
Theorem	19.35 1794	Theorem 19.35 of [Margaris] p. 90. This theorem is useful for moving an implication (in the form of the right-hand side) into the scope of a single existential quantifier. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
⊢ (∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
Theorem	19.35i 1795	Inference associated with 19.35 1794. (Contributed by NM, 21-Jun-1993.)
⊢ ∃𝑥(𝜑 → 𝜓)    ⇒   ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
Theorem	19.35ri 1796	Inference associated with 19.35 1794. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑥𝜑 → ∃𝑥𝜓)    ⇒   ⊢ ∃𝑥(𝜑 → 𝜓)
Theorem	19.25 1797	Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∃𝑦∃𝑥𝜓))
Theorem	19.30 1798	Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
⊢ (∀𝑥(𝜑 ∨ 𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
Theorem	19.43 1799	Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
Theorem	19.43OLD 1800	Obsolete proof of 19.43 1799. Do not delete as it is referenced on the mmrecent.html page and in conventions-label 26651. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
⊢ (∃𝑥(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))