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Theorem List for Metamath Proof Explorer - 1801-1900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theorem19.29r 1801 Variation of 19.29 1800. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.29r2 1802 Variation of 19.29r 1801 with double quantification. (Contributed by NM, 3-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
 
Theorem19.29x 1803 Variation of 19.29 1800 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
 
Theorem19.35 1804 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 1805 Inference associated with 19.35 1804. (Contributed by NM, 21-Jun-1993.)
𝑥(𝜑𝜓)       (∀𝑥𝜑 → ∃𝑥𝜓)
 
Theorem19.35ri 1806 Inference associated with 19.35 1804. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝜑 → ∃𝑥𝜓)       𝑥(𝜑𝜓)
 
Theorem19.25 1807 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑦𝑥(𝜑𝜓) → (∃𝑦𝑥𝜑 → ∃𝑦𝑥𝜓))
 
Theorem19.30 1808 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 1809 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 1810 Obsolete proof of 19.43 1809. Do not delete as it is referenced on the mmrecent.html page and in conventions-label 27243. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
 
Theorem19.33 1811 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 
Theorem19.33b 1812 The antecedent provides a condition implying the converse of 19.33 1811. (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)))
 
Theorem19.40-2 1813 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 1814 The antecedent provides a condition implying the converse of 19.40 1796. This is to 19.40 1796 what 19.33b 1812 is to 19.33 1811. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))
 
Theoremalbiim 1815 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
 
Theorem2albiim 1816 Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))
 
Theoremexintrbi 1817 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))
 
Theoremexintr 1818 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
 
Theoremalsyl 1819 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))
 
Theoremnfimt 1820 Closed form of nfim 1824 and curried (exported) form of nfimt2 1821. (Contributed by BJ, 20-Oct-2021.)
(Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))
 
Theoremnfimt2 1821 Closed form of nfim 1824 and uncurried (imported) form of nfimt 1820. (Contributed by BJ, 20-Oct-2021.)
((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜓) → Ⅎ𝑥(𝜑𝜓))
 
Theoremnfimd 1822 If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). Deduction form of nfimt 1820. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1709 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))
 
TheoremnfimdOLDOLD 1823 Obsolete proof of nfimd 1822 as of 3-Nov-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1709 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))
 
Theoremnfim 1824 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). Inference associated with nfimt 1820. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1709 changed. (Revised by Wolf Lammen, 17-Sep-2021.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnfand 1825 If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). (Contributed by Mario Carneiro, 7-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))
 
Theoremnf3and 1826 Deduction form of bound-variable hypothesis builder nf3an 1830. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑥𝜃)       (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))
 
Theoremnfan 1827 If 𝑥 is not free in 𝜑 and 𝜓, 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)
 
TheoremnfanOLD 1828 Obsolete proof of nfan 1827 as of 9-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 13-Jan-2018.) (New usage is discouraged.) (Proof modification is discouraged.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnfnan 1829 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 1830 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)
 
Theoremnfbid 1831 If in a context 𝑥 is not free in 𝜓 and 𝜒, 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 1832 If 𝑥 is not free in 𝜑 and 𝜓, 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 1833 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)
 
Theoremnf3or 1834 If 𝑥 is not free in 𝜑, 𝜓, and 𝜒, it is not free in (𝜑𝜓𝜒). (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑    &   𝑥𝜓    &   𝑥𝜒       𝑥(𝜑𝜓𝜒)
 
TheoremnfbiiOLD 1835 Obsolete proof of nfbii 1777 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
 
TheoremnfxfrOLD 1836 Obsolete proof of nfxfr 1778 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝑥𝜓       𝑥𝜑
 
TheoremnfxfrdOLD 1837 Obsolete proof of nfxfrd 1779 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   (𝜒 → Ⅎ𝑥𝜓)       (𝜒 → Ⅎ𝑥𝜑)
 
1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d
 
Axiomax-5 1838* Axiom of Distinctness. This axiom quantifies a variable over a formula in which it does not occur. Axiom C5 in [Megill] p. 444 (p. 11 of the preprint). Also appears as Axiom B6 (p. 75) of system S2 of [Tarski] p. 77 and Axiom C5-1 of [Monk2] p. 113.

(See comments in ax5ALT 34018 about the logical redundancy of ax-5 1838 in the presence of our obsolete axioms.)

This axiom essentially says that if 𝑥 does not occur in 𝜑, i.e. 𝜑 does not depend on 𝑥 in any way, then we can add the quantifier 𝑥 to 𝜑 with no further assumptions. By sp 2052, we can also remove the quantifier (unconditionally). (Contributed by NM, 10-Jan-1993.)

(𝜑 → ∀𝑥𝜑)
 
Theoremax5d 1839* ax-5 1838 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
(𝜑 → (𝜓 → ∀𝑥𝜓))
 
Theoremax5e 1840* A rephrasing of ax-5 1838 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)
 
Theoremax5ea 1841* If a formula holds for some value of a variable not occurring in it, then it holds for all values of that variable. (Contributed by BJ, 28-Dec-2020.)
(∃𝑥𝜑 → ∀𝑥𝜑)
 
Theoremnfv 1842* If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.) Definition change. (Revised by Wolf Lammen, 12-Sep-2021.)
𝑥𝜑
 
Theoremnfvd 1843* nfv 1842 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1822. (Contributed by Mario Carneiro, 6-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)
 
Theoremalimdv 1844* Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1737. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
Theoremeximdv 1845* Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1760. (Contributed by NM, 27-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theorem2alimdv 1846* Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1737. (Contributed by NM, 27-Apr-2004.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))
 
Theorem2eximdv 1847* Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1760. (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 → ∃𝑥𝑦𝜒))
 
Theoremalbidv 1848* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 
Theoremexbidv 1849* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
Theorem2albidv 1850* Formula-building rule for two universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 ↔ ∀𝑥𝑦𝜒))
 
Theorem2exbidv 1851* Formula-building rule for two existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))
 
Theorem3exbidv 1852* Formula-building rule for three existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))
 
Theorem4exbidv 1853* Formula-building rule for four existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))
 
Theoremalrimiv 1854* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2074 and 19.21v 1867. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝜓)
 
Theoremalrimivv 1855* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2074 and 19.21v 1867. (Contributed by NM, 31-Jul-1995.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝑦𝜓)
 
Theoremalrimdv 1856* Deduction form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2074 and 19.21v 1867. (Contributed by NM, 10-Feb-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
Theoremexlimiv 1857* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2079.

See exlimi 2085 for a more general version requiring more axioms.

This inference, along with its many variants such as rexlimdv 3028, is used to implement a metatheorem called "Rule C" that is given in many logic textbooks. See, for example, Rule C in [Mendelson] p. 81, Rule C in [Margaris] p. 40, or Rule C in Hirst and Hirst's A Primer for Logic and Proof p. 59 (PDF p. 65) at http://www.appstate.edu/~hirstjl/primer/hirst.pdf. In informal proofs, the statement "Let 𝐶 be an element such that..." almost always means an implicit application of Rule C.

In essence, Rule C states that if we can prove that some element 𝑥 exists satisfying a wff, i.e. 𝑥𝜑(𝑥) where 𝜑(𝑥) has 𝑥 free, then we can use 𝜑(𝐶) as a hypothesis for the proof where 𝐶 is a new (fictitious) constant not appearing previously in the proof, nor in any axioms used, nor in the theorem to be proved. The purpose of Rule C is to get rid of the existential quantifier.

We cannot do this in Metamath directly. Instead, we use the original 𝜑 (containing 𝑥) as an antecedent for the main part of the proof. We eventually arrive at (𝜑𝜓) where 𝜓 is the theorem to be proved and does not contain 𝑥. Then we apply exlimiv 1857 to arrive at (∃𝑥𝜑𝜓). Finally, we separately prove 𝑥𝜑 and detach it with modus ponens ax-mp 5 to arrive at the final theorem 𝜓. (Contributed by NM, 21-Jun-1993.) Remove dependencies on ax-6 1887 and ax-8 1991. (Revised by Wolf Lammen, 4-Dec-2017.)

(𝜑𝜓)       (∃𝑥𝜑𝜓)
 
Theoremexlimiiv 1858* Inference associated with exlimiv 1857. (Contributed by BJ, 19-Dec-2020.)
(𝜑𝜓)    &   𝑥𝜑       𝜓
 
Theoremexlimivv 1859* Inference form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2079. (Contributed by NM, 1-Aug-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑𝜓)
 
Theoremexlimdv 1860* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2079. (Contributed by NM, 27-Apr-1994.) Remove dependencies on ax-6 1887, ax-7 1934. (Revised by Wolf Lammen, 4-Dec-2017.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓𝜒))
 
Theoremexlimdvv 1861* Deduction form of Theorem 19.23 of [Margaris] p. 90, see 19.23 2079. (Contributed by NM, 31-Jul-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓𝜒))
 
Theoremexlimddv 1862* Existential elimination rule of natural deduction. (Contributed by Mario Carneiro, 15-Jun-2016.)
(𝜑 → ∃𝑥𝜓)    &   ((𝜑𝜓) → 𝜒)       (𝜑𝜒)
 
Theoremnexdv 1863* Deduction for generalization rule for negated wff. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)
 
TheoremnexdvOLD 1864* Obsolete proof of nexdv 1863 as of 10-Oct-2021. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 13-Jul-2020.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)
 
Theorem2ax5 1865* Quantification of two variables over a formula in which they do not occur. (Contributed by Alan Sare, 12-Apr-2011.)
(𝜑 → ∀𝑥𝑦𝜑)
 
Theoremstdpc5v 1866* Version of stdpc5 2075 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.) Revised to shorten 19.21v 1867. (Revised by Wolf Lammen, 12-Jul-2020.)
(∀𝑥(𝜑𝜓) → (𝜑 → ∀𝑥𝜓))
 
Theorem19.21v 1867* Version of 19.21 2074 with a dv condition, requiring fewer axioms.

Notational convention: We sometimes suffix with "v" the label of a theorem using a distinct variable ("dv") condition instead of a non-freeness hypothesis such as 𝑥𝜑. Conversely, we sometimes suffix with "f" the label of a theorem introducing such a non-freeness hypothesis ("f" stands for "not free in", see df-nf 1709) instead of a dv condition. For instance, 19.21v 1867 versus 19.21 2074 and vtoclf 3256 versus vtocl 3257. Note that "not free in" is less restrictive than "does not occur in." Note that the version with a dv condition is easily proved from the version with the corresponding non-freeness hypothesis, by using nfv 1842. However, the dv version can often be proved from fewer axioms. (Contributed by NM, 21-Jun-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 2-Jan-2020.) (Proof shortened by Wolf Lammen, 12-Jul-2020.)

(∀𝑥(𝜑𝜓) ↔ (𝜑 → ∀𝑥𝜓))
 
Theorem19.32v 1868* Version of 19.32 2100 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∀𝑥𝜓))
 
Theorem19.31v 1869* Version of 19.31 2101 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))
 
TheoremnfvOLD 1870* Obsolete proof of nfv 1842 as of 5-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
𝑥𝜑
 
TheoremnfvdOLD 1871* Obsolete proof of nfvd 1843 as of 6-Oct-2021. (Contributed by Mario Carneiro, 6-Oct-2016.) (New usage is discouraged.) (Proof modification is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)
 
TheoremnfdvOLD 1872* Obsolete proof of nf5dv 2024 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → (𝜓 → ∀𝑥𝜓))       (𝜑 → Ⅎ𝑥𝜓)
 
1.4.5  Equality predicate (continued)

The equality predicate was introduced above in wceq 1482 for use by df-tru 1485. See the comments in that section. In this section, we continue with the first "real" use of it.

 
Theoremweq 1873 Extend wff definition to include atomic formulas using the equality predicate.

(Instead of introducing weq 1873 as an axiomatic statement, as was done in an older version of this database, we introduce it by "proving" a special case of set theory's more general wceq 1482. This lets us avoid overloading the = connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically weq 1873 is considered to be a primitive syntax, even though here it is artificially "derived" from wceq 1482. Note: To see the proof steps of this syntax proof, type "show proof weq /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

wff 𝑥 = 𝑦
 
Theoremequs3 1874 Lemma used in proofs of substitution properties. (Contributed by NM, 10-May-1993.)
(∃𝑥(𝑥 = 𝑦𝜑) ↔ ¬ ∀𝑥(𝑥 = 𝑦 → ¬ 𝜑))
 
Theoremspeimfw 1875 Specialization, with additional weakening (compared to 19.2 1891) to allow bundling of 𝑥 and 𝑦. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Dec-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))
 
TheoremspeimfwALT 1876 Alternate proof of speimfw 1875 (longer compressed proof, but fewer essential steps). (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑 → ∃𝑥𝜓))
 
Theoremspimfw 1877 Specialization, with additional weakening (compared to sp 2052) to allow bundling of 𝑥 and 𝑦. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥 ¬ 𝑥 = 𝑦 → (∀𝑥𝜑𝜓))
 
Theoremax12i 1878 Inference that has ax-12 2046 (without 𝑦) as its conclusion. Uses only Tarski's FOL axiom schemes. The hypotheses may be eliminable without using ax-12 2046 in special cases. Proof similar to Lemma 16 of [Tarski] p. 70. (Contributed by NM, 20-May-2008.)
(𝑥 = 𝑦 → (𝜑𝜓))    &   (𝜓 → ∀𝑥𝜓)       (𝑥 = 𝑦 → (𝜑 → ∀𝑥(𝑥 = 𝑦𝜑)))
 
1.4.6  Define proper substitution
 
Syntaxwsb 1879 Extend wff definition to include proper substitution (read "the wff that results when 𝑦 is properly substituted for 𝑥 in wff 𝜑"). (Contributed by NM, 24-Jan-2006.)
wff [𝑦 / 𝑥]𝜑
 
Definitiondf-sb 1880 Define proper substitution. Remark 9.1 in [Megill] p. 447 (p. 15 of the preprint). For our notation, we use [𝑦 / 𝑥]𝜑 to mean "the wff that results from the proper substitution of 𝑦 for 𝑥 in the wff 𝜑." That is, 𝑦 properly replaces 𝑥. For example, [𝑥 / 𝑦]𝑧𝑦 is the same as 𝑧𝑥, as shown in elsb4 2434. We can also use [𝑦 / 𝑥]𝜑 in place of the "free for" side condition used in traditional predicate calculus; see, for example, stdpc4 2352.

Our notation was introduced in Haskell B. Curry's Foundations of Mathematical Logic (1977), p. 316 and is frequently used in textbooks of lambda calculus and combinatory logic. This notation improves the common but ambiguous notation, "𝜑(𝑦) is the wff that results when 𝑦 is properly substituted for 𝑥 in 𝜑(𝑥)." For example, if the original 𝜑(𝑥) is 𝑥 = 𝑦, then 𝜑(𝑦) is 𝑦 = 𝑦, from which we obtain that 𝜑(𝑥) is 𝑥 = 𝑥. So what exactly does 𝜑(𝑥) mean? Curry's notation solves this problem.

In most books, proper substitution has a somewhat complicated recursive definition with multiple cases based on the occurrences of free and bound variables in the wff. Instead, we use a single formula that is exactly equivalent and gives us a direct definition. We later prove that our definition has the properties we expect of proper substitution (see theorems sbequ 2375, sbcom2 2444 and sbid2v 2456).

Note that our definition is valid even when 𝑥 and 𝑦 are replaced with the same variable, as sbid 2113 shows. We achieve this by having 𝑥 free in the first conjunct and bound in the second. We can also achieve this by using a dummy variable, as the alternate definition dfsb7 2454 shows (which some logicians may prefer because it doesn't mix free and bound variables). Another version that mixes free and bound variables is dfsb3 2373. When 𝑥 and 𝑦 are distinct, we can express proper substitution with the simpler expressions of sb5 2429 and sb6 2428.

There are no restrictions on any of the variables, including what variables may occur in wff 𝜑. (Contributed by NM, 10-May-1993.)

([𝑦 / 𝑥]𝜑 ↔ ((𝑥 = 𝑦𝜑) ∧ ∃𝑥(𝑥 = 𝑦𝜑)))
 
Theoremsbequ2 1881 An equality theorem for substitution. (Contributed by NM, 16-May-1993.) (Proof shortened by Wolf Lammen, 25-Feb-2018.)
(𝑥 = 𝑦 → ([𝑦 / 𝑥]𝜑𝜑))
 
Theoremsb1 1882 One direction of a simplified definition of substitution. The converse requires either a dv condition (sb5 2429) or a non-freeness hypothesis (sb5f 2385). (Contributed by NM, 13-May-1993.)
([𝑦 / 𝑥]𝜑 → ∃𝑥(𝑥 = 𝑦𝜑))
 
Theoremspsbe 1883 A specialization theorem. (Contributed by NM, 29-Jun-1993.) (Proof shortened by Wolf Lammen, 3-May-2018.)
([𝑦 / 𝑥]𝜑 → ∃𝑥𝜑)
 
Theoremsbequ8 1884 Elimination of equality from antecedent after substitution. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 28-Jul-2018.)
([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥](𝑥 = 𝑦𝜑))
 
Theoremsbimi 1885 Infer substitution into antecedent and consequent of an implication. (Contributed by NM, 25-Jun-1998.)
(𝜑𝜓)       ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)
 
Theoremsbbii 1886 Infer substitution into both sides of a logical equivalence. (Contributed by NM, 14-May-1993.)
(𝜑𝜓)       ([𝑦 / 𝑥]𝜑 ↔ [𝑦 / 𝑥]𝜓)
 
1.4.7  Axiom scheme ax-6 (Existence)
 
Axiomax-6 1887 Axiom of Existence. One of the equality and substitution axioms of predicate calculus with equality. This axiom tells us is that at least one thing exists. In this form (not requiring that 𝑥 and 𝑦 be distinct) it was used in an axiom system of Tarski (see Axiom B7' in footnote 1 of [KalishMontague] p. 81.) It is equivalent to axiom scheme C10' in [Megill] p. 448 (p. 16 of the preprint); the equivalence is established by axc10 2251 and ax6fromc10 34007. A more convenient form of this axiom is ax6e 2249, which has additional remarks.

Raph Levien proved the independence of this axiom from the other logical axioms on 12-Apr-2005. See item 16 at http://us.metamath.org/award2003.html.

ax-6 1887 can be proved from the weaker version ax6v 1888 requiring that the variables be distinct; see theorem ax6 2250.

ax-6 1887 can also be proved from the Axiom of Separation (in the form that we use that axiom, where free variables are not universally quantified). See theorem ax6vsep 4783.

Except by ax6v 1888, this axiom should not be referenced directly. Instead, use theorem ax6 2250. (Contributed by NM, 10-Jan-1993.) (New usage is discouraged.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
Theoremax6v 1888* Axiom B7 of [Tarski] p. 75, which requires that 𝑥 and 𝑦 be distinct. This trivial proof is intended merely to weaken axiom ax-6 1887 by adding a distinct variable restriction ($d). From here on, ax-6 1887 should not be referenced directly by any other proof, so that theorem ax6 2250 will show that we can recover ax-6 1887 from this weaker version if it were an axiom (as it is in the case of Tarski).

Note: Introducing 𝑥, 𝑦 as a distinct variable group "out of the blue" with no apparent justification has puzzled some people, but it is perfectly sound. All we are doing is adding an additional prerequisite, similar to adding an unnecessary logical hypothesis, that results in a weakening of the theorem. This means that any future theorem that references ax6v 1888 must have a $d specified for the two variables that get substituted for 𝑥 and 𝑦. The $d does not propagate "backwards" i.e. it does not impose a requirement on ax-6 1887.

When possible, use of this theorem rather than ax6 2250 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 7-Aug-2015.)

¬ ∀𝑥 ¬ 𝑥 = 𝑦
 
Theoremax6ev 1889* At least one individual exists. Weaker version of ax6e 2249. When possible, use of this theorem rather than ax6e 2249 is preferred since its derivation is much shorter and requires fewer axioms. (Contributed by NM, 3-Aug-2017.)
𝑥 𝑥 = 𝑦
 
Theoremexiftru 1890 Rule of existential generalization, similar to universal generalization ax-gen 1721, but valid only if an individual exists. Its proof requires ax-6 1887 but the equality predicate does not occur in its statement. Some fundamental theorems of predicate logic can be proven from ax-gen 1721, ax-4 1736 and this theorem alone, not requiring ax-7 1934 or excessive distinct variable conditions. (Contributed by Wolf Lammen, 12-Nov-2017.) (Proof shortened by Wolf Lammen, 9-Dec-2017.)
𝜑       𝑥𝜑
 
Theorem19.2 1891 Theorem 19.2 of [Margaris] p. 89. This corresponds to the axiom (D) of modal logic. Note: This proof is very different from Margaris' because we only have Tarski's FOL axiom schemes available at this point. See the later 19.2g 2057 for a more conventional proof of a more general result, which uses additional axioms. The reverse implication is the defining property of effective non-freeness (see df-nf 1709). (Contributed by NM, 2-Aug-2017.) Remove dependency on ax-7 1934. (Revised by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑 → ∃𝑥𝜑)
 
Theorem19.2d 1892 Deduction associated with 19.2 1891. (Contributed by BJ, 12-May-2019.)
(𝜑 → ∀𝑥𝜓)       (𝜑 → ∃𝑥𝜓)
 
Theorem19.8w 1893 Weak version of 19.8a 2051 and instance of 19.2d 1892. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.) (Revised by BJ, 31-Mar-2021.)
(𝜑 → ∀𝑥𝜑)       (𝜑 → ∃𝑥𝜑)
 
Theorem19.8v 1894* Version of 19.8a 2051 with a dv condition, requiring fewer axioms. Converse of ax5e 1840. (Contributed by BJ, 12-Mar-2020.)
(𝜑 → ∃𝑥𝜑)
 
Theorem19.9v 1895* Version of 19.9 2071 with a dv condition, requiring fewer axioms. Any formula can be existentially quantified using a variable which it does not contain. See also 19.3v 1896. (Contributed by NM, 28-May-1995.) Remove dependency on ax-7 1934. (Revised by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)
 
Theorem19.3v 1896* Version of 19.3 2068 with a dv condition, requiring fewer axioms. Any formula can be universally quantified using a variable which it does not contain. See also 19.9v 1895. (Contributed by Anthony Hart, 13-Sep-2011.) Remove dependency on ax-7 1934. (Revised by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑𝜑)
 
Theoremspvw 1897* Version of sp 2052 when 𝑥 does not occur in 𝜑. Converse of ax-5 1838. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.) (Proof shortened by Wolf Lammen, 4-Dec-2017.)
(∀𝑥𝜑𝜑)
 
Theorem19.39 1898 Theorem 19.39 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∃𝑥𝜑 → ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.24 1899 Theorem 19.24 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 → ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.34 1900 Theorem 19.34 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
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