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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
1.4.3  Axiom scheme ax-4 (Quantified Implication)
 
Axiomax-4 1801 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1802 for labeling consistency. It should be used only by alim 1802. (Contributed by NM, 21-May-2008.) Use alim 1802 instead. (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 
Theoremalim 1802 Restatement of Axiom ax-4 1801, for labeling consistency. It should be the only theorem using ax-4 1801. (Contributed by NM, 10-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 
Theoremalimi 1803 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
(𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)
 
Theorem2alimi 1804 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓)
 
Theoremala1 1805 Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∀𝑥𝜑 → ∀𝑥(𝜓𝜑))
 
Theoremal2im 1806 Closed form of al2imi 1807. Version of alim 1802 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
 
Theoremal2imi 1807 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
Theoremalanimi 1808 Variant of al2imi 1807 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
((𝜑𝜓) → 𝜒)       ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)
 
Theoremalimdh 1809 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1802. (Contributed by NM, 4-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
Theoremalbi 1810 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
 
Theoremalbii 1811 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
(𝜑𝜓)       (∀𝑥𝜑 ↔ ∀𝑥𝜓)
 
Theorem2albii 1812 Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓)
 
Theoremsylgt 1813 Closed form of sylg 1814. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜓𝜒) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))
 
Theoremsylg 1814 A syllogism combined with generalization. Inference associated with sylgt 1813. General form of alrimih 1815. (Contributed by BJ, 4-Oct-2019.)
(𝜑 → ∀𝑥𝜓)    &   (𝜓𝜒)       (𝜑 → ∀𝑥𝜒)
 
Theoremalrimih 1815 Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2198 and 19.21h 2287. Instance of sylg 1814. (Contributed by NM, 9-Jan-1993.) (Revised by BJ, 31-Mar-2021.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)
 
Theoremhbxfrbi 1816 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2942 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜓 → ∀𝑥𝜓)       (𝜑 → ∀𝑥𝜑)
 
Theoremalex 1817 Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1772. See also the dual pair alnex 1773 / exnal 1818. Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
 
Theoremexnal 1818 Existential quantification of negation is equivalent to negation of universal quantification. Dual of alnex 1773. See also the dual pair df-ex 1772 / alex 1817. Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
 
Theorem2nalexn 1819 Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)
 
Theorem2exnaln 1820 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
 
Theorem2nexaln 1821 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
 
Theoremalimex 1822 An equivalence between an implication with a universally quantified consequent and an implication with an existentially quantified antecedent. 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 1774. (Contributed by BJ, 12-May-2019.)
((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))
 
Theoremaleximi 1823 A variant of al2imi 1807: instead of applying 𝑥 quantifiers to the final implication, replace them with 𝑥. A shorter proof is possible using nfa1 2146, sps 2174 and eximd 2207, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theoremalexbii 1824 Biconditional form of aleximi 1823. (Contributed by BJ, 16-Nov-2020.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
Theoremexim 1825 Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 10-Jan-1993.) (Proof shortened by Wolf Lammen, 4-Jul-2014.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥𝜓))
 
Theoremeximi 1826 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑𝜓)       (∃𝑥𝜑 → ∃𝑥𝜓)
 
Theorem2eximi 1827 Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓)
 
Theoremeximii 1828 Inference associated with eximi 1826. (Contributed by BJ, 3-Feb-2018.)
𝑥𝜑    &   (𝜑𝜓)       𝑥𝜓
 
Theoremexa1 1829 Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∃𝑥𝜑 → ∃𝑥(𝜓𝜑))
 
Theorem19.38 1830 Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1831 and 19.38b 1832. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2197. (Revised by Wolf Lammen, 2-Jan-2018.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 
Theorem19.38a 1831 Under a non-freeness hypothesis, the implication 19.38 1830 can be strengthened to an equivalence. See also 19.38b 1832. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)
(Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
 
Theorem19.38b 1832 Under a non-freeness hypothesis, the implication 19.38 1830 can be strengthened to an equivalence. See also 19.38a 1831. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)
(Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))
 
Theoremimnang 1833 Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))
 
Theoremalinexa 1834 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
 
Theoremexnalimn 1835 Existential quantification of a conjunction expressed with only primitive symbols (, ¬, ). (Contributed by NM, 10-May-1993.) State the most general instance. (Revised by BJ, 29-Sep-2019.)
(∃𝑥(𝜑𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓))
 
Theoremalexn 1836 A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
(∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
 
Theorem2exnexn 1837 Theorem *11.51 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) (Proof shortened by Wolf Lammen, 25-Sep-2014.)
(∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
 
Theoremexbi 1838 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
 
Theoremexbii 1839 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
(𝜑𝜓)       (∃𝑥𝜑 ↔ ∃𝑥𝜓)
 
Theorem2exbii 1840 Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓)
 
Theorem3exbii 1841 Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓)
 
Theoremnfbiit 1842 Equivalence theorem for the non-freeness predicate. Closed form of nfbii 1843. (Contributed by Giovanni Mascellani, 10-Apr-2018.) Reduce axiom usage. (Revised by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))
 
Theoremnfbii 1843 Equality theorem for the non-freeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
(𝜑𝜓)       (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
 
Theoremnfxfr 1844 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝜓)    &   𝑥𝜓       𝑥𝜑
 
Theoremnfxfrd 1845 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 24-Sep-2016.)
(𝜑𝜓)    &   (𝜒 → Ⅎ𝑥𝜓)       (𝜒 → Ⅎ𝑥𝜑)
 
Theoremnfnbi 1846 A variable is non-free in a proposition if and only if it is so in its negation. (Contributed by BJ, 6-May-2019.)
(Ⅎ𝑥𝜑 ↔ Ⅎ𝑥 ¬ 𝜑)
 
Theoremnfnt 1847 If a variable is non-free in a proposition, then it is non-free in its negation. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) df-nf 1776 changed. (Revised by Wolf Lammen, 4-Oct-2021.)
(Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)
 
Theoremnfn 1848 Inference associated with nfnt 1847. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑       𝑥 ¬ 𝜑
 
Theoremnfnd 1849 Deduction associated with nfnt 1847. (Contributed by Mario Carneiro, 24-Sep-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥 ¬ 𝜓)
 
Theoremexanali 1850 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(𝜑𝜓))
 
Theorem2exanali 1851 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))
 
Theoremexancom 1852 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
 
Theoremexan 1853 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, 6-Nov-2022.) Expand hypothesis. (Revised by Steven Nguyen, 19-Jun-2023.)
𝑥𝜑    &   𝜓       𝑥(𝜑𝜓)
 
TheoremexanOLD 1854 Obsolete proof of exan 1853 as of 19-Jun-2023. (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, 6-Nov-2022.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)
 
Theoremalrimdh 1855 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2198 and 19.21h 2287. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
Theoremeximdh 1856 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theoremnexdh 1857 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)
 
Theoremalbidh 1858 Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 
Theoremexbidh 1859 Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
Theoremexsimpl 1860 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∃𝑥(𝜑𝜓) → ∃𝑥𝜑)
 
Theoremexsimpr 1861 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∃𝑥(𝜑𝜓) → ∃𝑥𝜓)
 
Theorem19.26 1862 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(𝜑𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓))
 
Theorem19.26-2 1863 Theorem 19.26 1862 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓))
 
Theorem19.26-3an 1864 Theorem 19.26 1862 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
(∀𝑥(𝜑𝜓𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))
 
Theorem19.29 1865 Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1866. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.29r 1866 Variation of 19.29 1865. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.29r2 1867 Variation of 19.29r 1866 with double quantification. (Contributed by NM, 3-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
 
Theorem19.29x 1868 Variation of 19.29 1865 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
 
Theorem19.35 1869 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(𝜑𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓))
 
Theorem19.35i 1870 Inference associated with 19.35 1869. (Contributed by NM, 21-Jun-1993.)
𝑥(𝜑𝜓)       (∀𝑥𝜑 → ∃𝑥𝜓)
 
Theorem19.35ri 1871 Inference associated with 19.35 1869. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝜑 → ∃𝑥𝜓)       𝑥(𝜑𝜓)
 
Theorem19.25 1872 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑦𝑥(𝜑𝜓) → (∃𝑦𝑥𝜑 → ∃𝑦𝑥𝜓))
 
Theorem19.30 1873 Theorem 19.30 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ∨ ∃𝑥𝜓))
 
Theorem19.43 1874 Theorem 19.43 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) (Proof shortened by Wolf Lammen, 27-Jun-2014.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
 
Theorem19.43OLD 1875 Obsolete proof of 19.43 1874. Do not delete as it is referenced on the mmrecent.html 1874 page and in conventions-labels 28108. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
 
Theorem19.33 1876 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 
Theorem19.33b 1877 The antecedent provides a condition implying the converse of 19.33 1876. (Contributed by NM, 27-Mar-2004.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by Wolf Lammen, 5-Jul-2014.)
(¬ (∃𝑥𝜑 ∧ ∃𝑥𝜓) → (∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓)))
 
Theorem19.40 1878 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
(∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
 
Theorem19.40-2 1879 Theorem *11.42 in [WhiteheadRussell] p. 163. Theorem 19.40 of [Margaris] p. 90 with two quantifiers. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦(𝜑𝜓) → (∃𝑥𝑦𝜑 ∧ ∃𝑥𝑦𝜓))
 
Theorem19.40b 1880 The antecedent provides a condition implying the converse of 19.40 1878. This is to 19.40 1878 what 19.33b 1877 is to 19.33 1876. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))
 
Theoremalbiim 1881 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
 
Theorem2albiim 1882 Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))
 
Theoremexintrbi 1883 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))
 
Theoremexintr 1884 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.) (Proof shortened by BJ, 16-Sep-2022.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
 
Theoremalsyl 1885 Universally quantified and uncurried (imported) form of syllogism. Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜒)) → ∀𝑥(𝜑𝜒))
 
Theoremnfimd 1886 If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓𝜒). Deduction form of nfim 1888. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1776 changed. (Revised by Wolf Lammen, 18-Sep-2021.) Eliminate curried form of nfimt 1887. (Revised by Wolf Lammen, 10-Jul-2022.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))
 
Theoremnfimt 1887 Closed form of nfim 1888 and nfimd 1886. (Contributed by BJ, 20-Oct-2021.) Eliminate curried form, former name nfimt2. (Revised by Wolf Lammen, 6-Jul-2022.)
((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜓) → Ⅎ𝑥(𝜑𝜓))
 
Theoremnfim 1888 If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑𝜓). Inference associated with nfimt 1887. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1776 changed. (Revised by Wolf Lammen, 17-Sep-2021.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnfand 1889 If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))
 
Theoremnf3and 1890 Deduction form of bound-variable hypothesis builder nf3an 1893. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑥𝜃)       (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))
 
Theoremnfan 1891 If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 9-Oct-2021.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnfnan 1892 If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑𝜓). (Contributed by Scott Fenton, 2-Jan-2018.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnf3an 1893 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, then it is not free in (𝜑𝜓𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)
 
Theoremnfbid 1894 If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))
 
Theoremnfbi 1895 If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑𝜓). (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnfor 1896 If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnf3or 1897 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, then it is not free in (𝜑𝜓𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)
 
1.4.3.1  The empty domain of discourse

This database develops mathematics from first-order logic, which has only nonempty models. Before stating axioms excluding the empty model (typically, ax-6 1961 in logic and ax-nul 5202 in set theory), we state in this short subsection a few results relative to the empty domain, which we characterize by the assumption ¬ ∃𝑥. As expected, on the empty domain, every universally quantified formula is true (emptyal 1900) and every existential formula is false (emptyex 1899), and every variable is effectively nonfree in any formula (emptynf 1901).

 
Theoremempty 1898 Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.)
(¬ ∃𝑥⊤ ↔ ∀𝑥⊥)
 
Theorememptyex 1899 On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.)
(¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑)
 
Theorememptyal 1900 On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023.)
(¬ ∃𝑥⊤ → ∀𝑥𝜑)
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