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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theoremmpgbir 1801 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

Theoremnex 1802 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
¬ 𝜑        ¬ ∃𝑥𝜑

Theoremnfth 1803 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
𝜑       𝑥𝜑

Theoremnfnth 1804 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
¬ 𝜑       𝑥𝜑

Theoremhbth 1805 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.) This hb* idiom is generally being replaced by the nf* idiom (see nfth 1803), but keeps its interest in some cases. (Revised by BJ, 23-Sep-2022.)

𝜑       (𝜑 → ∀𝑥𝜑)

Theoremnftru 1806 The true constant has no free variables. (This can also be proven in one step with nfv 1916, but this proof does not use ax-5 1912.) (Contributed by Mario Carneiro, 6-Oct-2016.)
𝑥

Theoremnffal 1807 The false constant has no free variables (see nftru 1806). (Contributed by BJ, 6-May-2019.)
𝑥

Theoremsptruw 1808 Version of sp 2184 when 𝜑 is true. Instance of a1i 11. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.)
𝜑       (∀𝑥𝜑𝜑)

Theoremaltru 1809 For all sets, is true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥

Theoremalfal 1810 For all sets, ¬ ⊥ is true. (Contributed by Anthony Hart, 13-Sep-2011.)
𝑥 ¬ ⊥

1.4.3  Axiom scheme ax-4 (Quantified Implication)

Axiomax-4 1811 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1812 for labeling consistency. It should be used only by alim 1812. (Contributed by NM, 21-May-2008.) Use alim 1812 instead. (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremalim 1812 Restatement of Axiom ax-4 1811, for labeling consistency. It should be the only theorem using ax-4 1811. (Contributed by NM, 10-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))

Theoremalimi 1813 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
(𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)

Theorem2alimi 1814 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓)

Theoremala1 1815 Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∀𝑥𝜑 → ∀𝑥(𝜓𝜑))

Theoremal2im 1816 Closed form of al2imi 1817. Version of alim 1812 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))

Theoremal2imi 1817 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalanimi 1818 Variant of al2imi 1817 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
((𝜑𝜓) → 𝜒)       ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)

Theoremalimdh 1819 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1812. (Contributed by NM, 4-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremalbi 1820 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))

Theoremalbii 1821 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
(𝜑𝜓)       (∀𝑥𝜑 ↔ ∀𝑥𝜓)

Theorem2albii 1822 Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓)

Theoremsylgt 1823 Closed form of sylg 1824. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜓𝜒) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))

Theoremsylg 1824 A syllogism combined with generalization. Inference associated with sylgt 1823. General form of alrimih 1825. (Contributed by BJ, 4-Oct-2019.)
(𝜑 → ∀𝑥𝜓)    &   (𝜓𝜒)       (𝜑 → ∀𝑥𝜒)

Theoremalrimih 1825 Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2209 and 19.21h 2297. Instance of sylg 1824. (Contributed by NM, 9-Jan-1993.) (Revised by BJ, 31-Mar-2021.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

Theoremhbxfrbi 1826 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2945 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜓 → ∀𝑥𝜓)       (𝜑 → ∀𝑥𝜑)

Theoremalex 1827 Universal quantifier in terms of existential quantifier and negation. Dual of df-ex 1782. See also the dual pair alnex 1783 / exnal 1828. Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)

Theoremexnal 1828 Existential quantification of negation is equivalent to negation of universal quantification. Dual of alnex 1783. See also the dual pair df-ex 1782 / alex 1827. Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)

Theorem2nalexn 1829 Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)

Theorem2exnaln 1830 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)

Theorem2nexaln 1831 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)

Theoremalimex 1832 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 1784. (Contributed by BJ, 12-May-2019.)
((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))

Theoremaleximi 1833 A variant of al2imi 1817: instead of applying 𝑥 quantifiers to the final implication, replace them with 𝑥. A shorter proof is possible using nfa1 2156, sps 2186 and eximd 2218, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremalexbii 1834 Biconditional form of aleximi 1833. (Contributed by BJ, 16-Nov-2020.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremexim 1835 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 1836 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑𝜓)       (∃𝑥𝜑 → ∃𝑥𝜓)

Theorem2eximi 1837 Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓)

Theoremeximii 1838 Inference associated with eximi 1836. (Contributed by BJ, 3-Feb-2018.)
𝑥𝜑    &   (𝜑𝜓)       𝑥𝜓

Theoremexa1 1839 Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∃𝑥𝜑 → ∃𝑥(𝜓𝜑))

Theorem19.38 1840 Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1841 and 19.38b 1842. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2208. (Revised by Wolf Lammen, 2-Jan-2018.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.38a 1841 Under a non-freeness hypothesis, the implication 19.38 1840 can be strengthened to an equivalence. See also 19.38b 1842. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)
(Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Theorem19.38b 1842 Under a non-freeness hypothesis, the implication 19.38 1840 can be strengthened to an equivalence. See also 19.38a 1841. (Contributed by BJ, 3-Nov-2021.) (Proof shortened by Wolf Lammen, 9-Jul-2022.)
(Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Theoremimnang 1843 Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))

Theoremalinexa 1844 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))

Theoremexnalimn 1845 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 1846 A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
(∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥𝑦𝜑)

Theorem2exnexn 1847 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 1848 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Theoremexbii 1849 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
(𝜑𝜓)       (∃𝑥𝜑 ↔ ∃𝑥𝜓)

Theorem2exbii 1850 Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓)

Theorem3exbii 1851 Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓)

Theoremnfbiit 1852 Equivalence theorem for the non-freeness predicate. Closed form of nfbii 1853. (Contributed by Giovanni Mascellani, 10-Apr-2018.) Reduce axiom usage. (Revised by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Theoremnfbii 1853 Equality theorem for the non-freeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
(𝜑𝜓)       (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Theoremnfxfr 1854 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝜓)    &   𝑥𝜓       𝑥𝜑

Theoremnfxfrd 1855 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 1856 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 1857 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 1786 changed. (Revised by Wolf Lammen, 4-Oct-2021.)
(Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Theoremnfn 1858 Inference associated with nfnt 1857. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑       𝑥 ¬ 𝜑

Theoremnfnd 1859 Deduction associated with nfnt 1857. (Contributed by Mario Carneiro, 24-Sep-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥 ¬ 𝜓)

Theoremexanali 1860 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 1861 Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥𝑦(𝜑𝜓))

Theoremexancom 1862 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))

Theoremexan 1863 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 1864 Obsolete proof of exan 1863 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 1865 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2209 and 19.21h 2297. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremeximdh 1866 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremnexdh 1867 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theoremalbidh 1868 Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbidh 1869 Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremexsimpl 1870 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 1871 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 1872 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 1873 Theorem 19.26 1872 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓))

Theorem19.26-3an 1874 Theorem 19.26 1872 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
(∀𝑥(𝜑𝜓𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))

Theorem19.29 1875 Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1876. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.29r 1876 Variation of 19.29 1875. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.29r2 1877 Variation of 19.29r 1876 with double quantification. (Contributed by NM, 3-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

Theorem19.29x 1878 Variation of 19.29 1875 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

Theorem19.35 1879 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 1880 Inference associated with 19.35 1879. (Contributed by NM, 21-Jun-1993.)
𝑥(𝜑𝜓)       (∀𝑥𝜑 → ∃𝑥𝜓)

Theorem19.35ri 1881 Inference associated with 19.35 1879. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝜑 → ∃𝑥𝜓)       𝑥(𝜑𝜓)

Theorem19.25 1882 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑦𝑥(𝜑𝜓) → (∃𝑦𝑥𝜑 → ∃𝑦𝑥𝜓))

Theorem19.30 1883 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 1884 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 1885 Obsolete proof of 19.43 1884. Do not delete as it is referenced on the mmrecent.html 1884 page and in conventions-labels 28175. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Theorem19.33 1886 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.33b 1887 The antecedent provides a condition implying the converse of 19.33 1886. (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 1888 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
(∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Theorem19.40-2 1889 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 1890 The antecedent provides a condition implying the converse of 19.40 1888. This is to 19.40 1888 what 19.33b 1887 is to 19.33 1886. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))

Theoremalbiim 1891 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))

Theorem2albiim 1892 Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))

Theoremexintrbi 1893 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))

Theoremexintr 1894 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 1895 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 1896 If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓𝜒). Deduction form of nfim 1898. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1786 changed. (Revised by Wolf Lammen, 18-Sep-2021.) Eliminate curried form of nfimt 1897. (Revised by Wolf Lammen, 10-Jul-2022.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremnfimt 1897 Closed form of nfim 1898 and nfimd 1896. (Contributed by BJ, 20-Oct-2021.) Eliminate curried form, former name nfimt2. (Revised by Wolf Lammen, 6-Jul-2022.)
((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜓) → Ⅎ𝑥(𝜑𝜓))

Theoremnfim 1898 If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑𝜓). Inference associated with nfimt 1897. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1786 changed. (Revised by Wolf Lammen, 17-Sep-2021.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremnfand 1899 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 1900 Deduction form of bound-variable hypothesis builder nf3an 1903. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑥𝜃)       (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))

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