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Type | Label | Description |
---|---|---|
Statement | ||
Axiom | ax-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(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | alim 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(𝜑 → 𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓)) | ||
Theorem | alimi 1803 | Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∀𝑥𝜓) | ||
Theorem | 2alimi 1804 | Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥∀𝑦𝜓) | ||
Theorem | ala1 1805 | Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.) |
⊢ (∀𝑥𝜑 → ∀𝑥(𝜓 → 𝜑)) | ||
Theorem | al2im 1806 | Closed form of al2imi 1807. Version of alim 1802 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.) |
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))) | ||
Theorem | al2imi 1807 | Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.) |
⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) | ||
Theorem | alanimi 1808 | Variant of al2imi 1807 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.) |
⊢ ((𝜑 ∧ 𝜓) → 𝜒) ⇒ ⊢ ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒) | ||
Theorem | alimdh 1809 | Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1802. (Contributed by NM, 4-Jan-2002.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)) | ||
Theorem | albi 1810 | Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓)) | ||
Theorem | albii 1811 | Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∀𝑥𝜑 ↔ ∀𝑥𝜓) | ||
Theorem | 2albii 1812 | 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 1813 | Closed form of sylg 1814. (Contributed by BJ, 2-May-2019.) |
⊢ (∀𝑥(𝜓 → 𝜒) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒))) | ||
Theorem | sylg 1814 | A syllogism combined with generalization. Inference associated with sylgt 1813. General form of alrimih 1815. (Contributed by BJ, 4-Oct-2019.) |
⊢ (𝜑 → ∀𝑥𝜓) & ⊢ (𝜓 → 𝜒) ⇒ ⊢ (𝜑 → ∀𝑥𝜒) | ||
Theorem | alrimih 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) & ⊢ (𝜑 → 𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜓) | ||
Theorem | hbxfrbi 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.) |
⊢ (𝜑 ↔ 𝜓) & ⊢ (𝜓 → ∀𝑥𝜓) ⇒ ⊢ (𝜑 → ∀𝑥𝜑) | ||
Theorem | alex 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) | ||
Theorem | exnal 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) | ||
Theorem | 2nalexn 1819 | Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (¬ ∀𝑥∀𝑦𝜑 ↔ ∃𝑥∃𝑦 ¬ 𝜑) | ||
Theorem | 2exnaln 1820 | Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (∃𝑥∃𝑦𝜑 ↔ ¬ ∀𝑥∀𝑦 ¬ 𝜑) | ||
Theorem | 2nexaln 1821 | Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (¬ ∃𝑥∃𝑦𝜑 ↔ ∀𝑥∀𝑦 ¬ 𝜑) | ||
Theorem | alimex 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)) | ||
Theorem | aleximi 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 → (𝜓 → 𝜒)) ⇒ ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒)) | ||
Theorem | alexbii 1824 | Biconditional form of aleximi 1823. (Contributed by BJ, 16-Nov-2020.) |
⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) | ||
Theorem | exim 1825 | 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 1826 | Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥𝜑 → ∃𝑥𝜓) | ||
Theorem | 2eximi 1827 | Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.) |
⊢ (𝜑 → 𝜓) ⇒ ⊢ (∃𝑥∃𝑦𝜑 → ∃𝑥∃𝑦𝜓) | ||
Theorem | eximii 1828 | Inference associated with eximi 1826. (Contributed by BJ, 3-Feb-2018.) |
⊢ ∃𝑥𝜑 & ⊢ (𝜑 → 𝜓) ⇒ ⊢ ∃𝑥𝜓 | ||
Theorem | exa1 1829 | Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.) |
⊢ (∃𝑥𝜑 → ∃𝑥(𝜓 → 𝜑)) | ||
Theorem | 19.38 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(𝜑 → 𝜓)) | ||
Theorem | 19.38a 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(𝜑 → 𝜓))) | ||
Theorem | 19.38b 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(𝜑 → 𝜓))) | ||
Theorem | imnang 1833 | Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑 ∧ 𝜓)) | ||
Theorem | alinexa 1834 | A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.) |
⊢ (∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | exnalimn 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(𝜑 ∧ 𝜓) ↔ ¬ ∀𝑥(𝜑 → ¬ 𝜓)) | ||
Theorem | alexn 1836 | A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.) |
⊢ (∀𝑥∃𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥∀𝑦𝜑) | ||
Theorem | 2exnexn 1837 | 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 1838 | Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓)) | ||
Theorem | exbii 1839 | Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (∃𝑥𝜑 ↔ ∃𝑥𝜓) | ||
Theorem | 2exbii 1840 | 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 1841 | 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 | nfbiit 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(𝜑 ↔ 𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)) | ||
Theorem | nfbii 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.) |
⊢ (𝜑 ↔ 𝜓) ⇒ ⊢ (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓) | ||
Theorem | nfxfr 1844 | 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 1845 | 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 | nfnbi 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) | ||
Theorem | nfnt 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) | ||
Theorem | nfn 1848 | Inference associated with nfnt 1847. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1776 changed. (Revised by Wolf Lammen, 18-Sep-2021.) |
⊢ Ⅎ𝑥𝜑 ⇒ ⊢ Ⅎ𝑥 ¬ 𝜑 | ||
Theorem | nfnd 1849 | Deduction associated with nfnt 1847. (Contributed by Mario Carneiro, 24-Sep-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) ⇒ ⊢ (𝜑 → Ⅎ𝑥 ¬ 𝜓) | ||
Theorem | exanali 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(𝜑 → 𝜓)) | ||
Theorem | 2exanali 1851 | Theorem *11.521 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.) |
⊢ (¬ ∃𝑥∃𝑦(𝜑 ∧ ¬ 𝜓) ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | ||
Theorem | exancom 1852 | Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.) |
⊢ (∃𝑥(𝜑 ∧ 𝜓) ↔ ∃𝑥(𝜓 ∧ 𝜑)) | ||
Theorem | exan 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(𝜑 ∧ 𝜓) | ||
Theorem | exanOLD 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.) |
⊢ (∃𝑥𝜑 ∧ 𝜓) ⇒ ⊢ ∃𝑥(𝜑 ∧ 𝜓) | ||
Theorem | alrimdh 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 → (𝜓 → 𝜒)) ⇒ ⊢ (𝜑 → (𝜓 → ∀𝑥𝜒)) | ||
Theorem | eximdh 1856 | 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 1857 | Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → ¬ 𝜓) ⇒ ⊢ (𝜑 → ¬ ∃𝑥𝜓) | ||
Theorem | albidh 1858 | Formula-building rule for universal quantifier (deduction form). (Contributed by NM, 26-May-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒)) | ||
Theorem | exbidh 1859 | Formula-building rule for existential quantifier (deduction form). (Contributed by NM, 26-May-1993.) |
⊢ (𝜑 → ∀𝑥𝜑) & ⊢ (𝜑 → (𝜓 ↔ 𝜒)) ⇒ ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒)) | ||
Theorem | exsimpl 1860 | 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 1861 | 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 | 19.26 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(𝜑 ∧ 𝜓) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓)) | ||
Theorem | 19.26-2 1863 | Theorem 19.26 1862 with two quantifiers. (Contributed by NM, 3-Feb-2005.) |
⊢ (∀𝑥∀𝑦(𝜑 ∧ 𝜓) ↔ (∀𝑥∀𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓)) | ||
Theorem | 19.26-3an 1864 | Theorem 19.26 1862 with triple conjunction. (Contributed by NM, 13-Sep-2011.) |
⊢ (∀𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒)) | ||
Theorem | 19.29 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(𝜑 ∧ 𝜓)) | ||
Theorem | 19.29r 1866 | Variation of 19.29 1865. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.) |
⊢ ((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑 ∧ 𝜓)) | ||
Theorem | 19.29r2 1867 | Variation of 19.29r 1866 with double quantification. (Contributed by NM, 3-Feb-2005.) |
⊢ ((∃𝑥∃𝑦𝜑 ∧ ∀𝑥∀𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓)) | ||
Theorem | 19.29x 1868 | Variation of 19.29 1865 with mixed quantification. (Contributed by NM, 11-Feb-2005.) |
⊢ ((∃𝑥∀𝑦𝜑 ∧ ∀𝑥∃𝑦𝜓) → ∃𝑥∃𝑦(𝜑 ∧ 𝜓)) | ||
Theorem | 19.35 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(𝜑 → 𝜓) ↔ (∀𝑥𝜑 → ∃𝑥𝜓)) | ||
Theorem | 19.35i 1870 | Inference associated with 19.35 1869. (Contributed by NM, 21-Jun-1993.) |
⊢ ∃𝑥(𝜑 → 𝜓) ⇒ ⊢ (∀𝑥𝜑 → ∃𝑥𝜓) | ||
Theorem | 19.35ri 1871 | Inference associated with 19.35 1869. (Contributed by NM, 12-Mar-1993.) |
⊢ (∀𝑥𝜑 → ∃𝑥𝜓) ⇒ ⊢ ∃𝑥(𝜑 → 𝜓) | ||
Theorem | 19.25 1872 | Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ (∀𝑦∃𝑥(𝜑 → 𝜓) → (∃𝑦∀𝑥𝜑 → ∃𝑦∃𝑥𝜓)) | ||
Theorem | 19.30 1873 | 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 1874 | 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 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(𝜑 ∨ 𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓)) | ||
Theorem | 19.33 1876 | Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) |
⊢ ((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑 ∨ 𝜓)) | ||
Theorem | 19.33b 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(𝜑 ∨ 𝜓) ↔ (∀𝑥𝜑 ∨ ∀𝑥𝜓))) | ||
Theorem | 19.40 1878 | Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.) |
⊢ (∃𝑥(𝜑 ∧ 𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓)) | ||
Theorem | 19.40-2 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.) |
⊢ (∃𝑥∃𝑦(𝜑 ∧ 𝜓) → (∃𝑥∃𝑦𝜑 ∧ ∃𝑥∃𝑦𝜓)) | ||
Theorem | 19.40b 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(𝜑 ∧ 𝜓))) | ||
Theorem | albiim 1881 | Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.) |
⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜑))) | ||
Theorem | 2albiim 1882 | Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.) |
⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) | ||
Theorem | exintrbi 1883 | Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.) |
⊢ (∀𝑥(𝜑 → 𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑 ∧ 𝜓))) | ||
Theorem | exintr 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(𝜑 → 𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑 ∧ 𝜓))) | ||
Theorem | alsyl 1885 | Universally quantified and uncurried (imported) form of syllogism. Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.) |
⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∀𝑥(𝜓 → 𝜒)) → ∀𝑥(𝜑 → 𝜒)) | ||
Theorem | nfimd 1886 | If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 → 𝜒). 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(𝜓 → 𝜒)) | ||
Theorem | nfimt 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(𝜑 → 𝜓)) | ||
Theorem | nfim 1888 | If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 → 𝜓). 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.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 → 𝜓) | ||
Theorem | nfand 1889 | If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 7-Oct-2016.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 ∧ 𝜒)) | ||
Theorem | nf3and 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(𝜓 ∧ 𝜒 ∧ 𝜃)) | ||
Theorem | nfan 1891 | If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ∧ 𝜓). (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (Proof shortened by Wolf Lammen, 9-Oct-2021.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓) | ||
Theorem | nfnan 1892 | If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ⊼ 𝜓). (Contributed by Scott Fenton, 2-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ⊼ 𝜓) | ||
Theorem | nf3an 1893 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, then it is not free in (𝜑 ∧ 𝜓 ∧ 𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓 ∧ 𝜒) | ||
Theorem | nfbid 1894 | If in a context 𝑥 is not free in 𝜓 and 𝜒, then it is not free in (𝜓 ↔ 𝜒). (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 29-Dec-2017.) |
⊢ (𝜑 → Ⅎ𝑥𝜓) & ⊢ (𝜑 → Ⅎ𝑥𝜒) ⇒ ⊢ (𝜑 → Ⅎ𝑥(𝜓 ↔ 𝜒)) | ||
Theorem | nfbi 1895 | If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ↔ 𝜓). (Contributed by NM, 26-May-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ↔ 𝜓) | ||
Theorem | nfor 1896 | If 𝑥 is not free in 𝜑 and 𝜓, then it is not free in (𝜑 ∨ 𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 ⇒ ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓) | ||
Theorem | nf3or 1897 | If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, then it is not free in (𝜑 ∨ 𝜓 ∨ 𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.) |
⊢ Ⅎ𝑥𝜑 & ⊢ Ⅎ𝑥𝜓 & ⊢ Ⅎ𝑥𝜒 ⇒ ⊢ Ⅎ𝑥(𝜑 ∨ 𝜓 ∨ 𝜒) | ||
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). | ||
Theorem | empty 1898 | Two characterizations of the empty domain. (Contributed by Gérard Lang, 5-Feb-2024.) |
⊢ (¬ ∃𝑥⊤ ↔ ∀𝑥⊥) | ||
Theorem | emptyex 1899 | On the empty domain, any existentially quantified formula is false. (Contributed by Wolf Lammen, 21-Jan-2024.) |
⊢ (¬ ∃𝑥⊤ → ¬ ∃𝑥𝜑) | ||
Theorem | emptyal 1900 | On the empty domain, any universally quantified formula is true. (Contributed by Wolf Lammen, 12-Mar-2023.) |
⊢ (¬ ∃𝑥⊤ → ∀𝑥𝜑) |
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