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Theorem List for Metamath Proof Explorer - 1701-1800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremgen2 1701 Generalization applied twice. (Contributed by NM, 30-Apr-1998.)
𝜑       𝑥𝑦𝜑
 
Theoremmpg 1702 Modus ponens combined with generalization. (Contributed by NM, 24-May-1994.)
(∀𝑥𝜑𝜓)    &   𝜑       𝜓
 
Theoremmpgbi 1703 Modus ponens on biconditional combined with generalization. (Contributed by NM, 24-May-1994.) (Proof shortened by Stefan Allan, 28-Oct-2008.)
(∀𝑥𝜑𝜓)    &   𝜑       𝜓
 
Theoremmpgbir 1704 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
 
Theoremnfi 1705 Deduce that 𝑥 is not free in 𝜑 from the definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑 → ∀𝑥𝜑)       𝑥𝜑
 
Theoremhbth 1706 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)
 
Theoremnfth 1707 No variable is (effectively) free in a theorem. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝜑       𝑥𝜑
 
Theoremnftru 1708 The true constant has no free variables. (This can also be proven in one step with nfv 1796, but this proof does not use ax-5 1793.) (Contributed by Mario Carneiro, 6-Oct-2016.)
𝑥
 
Theoremnex 1709 Generalization rule for negated wff. (Contributed by NM, 18-May-1994.)
¬ 𝜑        ¬ ∃𝑥𝜑
 
Theoremnfnth 1710 No variable is (effectively) free in a non-theorem. (Contributed by Mario Carneiro, 6-Dec-2016.)
¬ 𝜑       𝑥𝜑
 
Theoremnffal 1711 The false constant has no free variables (see nftru 1708). (Contributed by BJ, 6-May-2019.)
𝑥
 
Theoremsptruw 1712 Version of sp 1990 when 𝜑 is true. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.)
𝜑       (∀𝑥𝜑𝜑)
 
1.4.3  Axiom scheme ax-4 (Quantified Implication)
 
Axiomax-4 1713 Axiom of Quantified Implication. Axiom C4 of [Monk2] p. 105 and Theorem 19.20 of [Margaris] p. 90. It is restated as alim 1714 for labeling consistency. It should be used only by alim 1714. (Contributed by NM, 21-May-2008.) Use alim 1714 instead. (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 
Theoremalim 1714 Restatement of Axiom ax-4 1713, for labeling consistency. It should be the only theorem using ax-4 1713. (Contributed by NM, 10-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 → ∀𝑥𝜓))
 
Theoremalimi 1715 Inference quantifying both antecedent and consequent. (Contributed by NM, 5-Jan-1993.)
(𝜑𝜓)       (∀𝑥𝜑 → ∀𝑥𝜓)
 
Theorem2alimi 1716 Inference doubly quantifying both antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 → ∀𝑥𝑦𝜓)
 
Theoremal2im 1717 Closed form of al2imi 1718. Version of alim 1714 for a nested implication. (Contributed by Alan Sare, 31-Dec-2011.)
(∀𝑥(𝜑 → (𝜓𝜒)) → (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒)))
 
Theoremal2imi 1718 Inference quantifying antecedent, nested antecedent, and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
Theoremalanimi 1719 Variant of al2imi 1718 with conjunctive antecedent. (Contributed by Andrew Salmon, 8-Jun-2011.)
((𝜑𝜓) → 𝜒)       ((∀𝑥𝜑 ∧ ∀𝑥𝜓) → ∀𝑥𝜒)
 
Theoremalimdh 1720 Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1714. (Contributed by NM, 4-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
Theoremalbi 1721 Theorem 19.15 of [Margaris] p. 90. (Contributed by NM, 24-Jan-1993.)
(∀𝑥(𝜑𝜓) → (∀𝑥𝜑 ↔ ∀𝑥𝜓))
 
Theoremalbii 1722 Inference adding universal quantifier to both sides of an equivalence. (Contributed by NM, 7-Aug-1994.)
(𝜑𝜓)       (∀𝑥𝜑 ↔ ∀𝑥𝜓)
 
Theorem2albii 1723 Inference adding two universal quantifiers to both sides of an equivalence. (Contributed by NM, 9-Mar-1997.)
(𝜑𝜓)       (∀𝑥𝑦𝜑 ↔ ∀𝑥𝑦𝜓)
 
Theoremsylgt 1724 Closed form of sylg 1725. (Contributed by BJ, 2-May-2019.)
(∀𝑥(𝜓𝜒) → ((𝜑 → ∀𝑥𝜓) → (𝜑 → ∀𝑥𝜒)))
 
Theoremsylg 1725 A syllogism combined with generalization. Inference associated with sylgt 1724. General form of alrimih 1726. (Contributed by BJ, 4-Oct-2019.)
(𝜑 → ∀𝑥𝜓)    &   (𝜓𝜒)       (𝜑 → ∀𝑥𝜒)
 
Theoremalrimih 1726 Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2036 and 19.21h 2038. Instance of sylg 1725. (Contributed by NM, 9-Jan-1993.) (Revised by BJ, 31-Mar-2021.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑𝜓)       (𝜑 → ∀𝑥𝜓)
 
Theoremhbxfrbi 1727 A utility lemma to transfer a bound-variable hypothesis builder into a definition. See hbxfreq 2621 for equality version. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.)
(𝜑𝜓)    &   (𝜓 → ∀𝑥𝜓)       (𝜑 → ∀𝑥𝜑)
 
Theoremnfbii 1728 Equality theorem for not-free. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝜓)       (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)
 
Theoremnfxfr 1729 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝜓)    &   𝑥𝜓       𝑥𝜑
 
Theoremnfxfrd 1730 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)
 
Theoremalex 1731 Theorem 19.6 of [Margaris] p. 89. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝜑 ↔ ¬ ∃𝑥 ¬ 𝜑)
 
Theoremexnal 1732 Theorem 19.14 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∃𝑥 ¬ 𝜑 ↔ ¬ ∀𝑥𝜑)
 
Theorem2nalexn 1733 Part of theorem *11.5 in [WhiteheadRussell] p. 164. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∀𝑥𝑦𝜑 ↔ ∃𝑥𝑦 ¬ 𝜑)
 
Theorem2exnaln 1734 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)
 
Theorem2nexaln 1735 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)
 
Theoremalimex 1736 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 1697. (Contributed by BJ, 12-May-2019.)
((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))
 
Theoremaleximi 1737 A variant of al2imi 1718: instead of applying 𝑥 quantifiers to the final implication, replace them with 𝑥. A shorter proof is possible using nfa1 2027, sps 1996 and eximd 2013, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theoremalexbii 1738 Biconditional form of aleximi 1737. (Contributed by BJ, 16-Nov-2020.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
Theoremexim 1739 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 1740 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑𝜓)       (∃𝑥𝜑 → ∃𝑥𝜓)
 
Theorem2eximi 1741 Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓)
 
Theoremeximii 1742 Inference associated with eximi 1740. (Contributed by BJ, 3-Feb-2018.)
𝑥𝜑    &   (𝜑𝜓)       𝑥𝜓
 
Theoremala1 1743 Add an antecedent in a universally quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∀𝑥𝜑 → ∀𝑥(𝜓𝜑))
 
Theoremexa1 1744 Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∃𝑥𝜑 → ∃𝑥(𝜓𝜑))
 
Theorem19.38 1745 Theorem 19.38 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2035. (Revised by Wolf Lammen, 2-Jan-2018.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 
Theoremimnang 1746 Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))
 
Theoremalinexa 1747 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))
 
Theoremalexn 1748 A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
(∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥𝑦𝜑)
 
Theorem2exnexn 1749 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 1750 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
 
TheoremexbiOLD 1751 Obsolete proof of exbi 1750 as of 16-Nov-2020. (Contributed by NM, 12-Mar-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))
 
Theoremexbii 1752 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
(𝜑𝜓)       (∃𝑥𝜑 ↔ ∃𝑥𝜓)
 
Theorem2exbii 1753 Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓)
 
Theorem3exbii 1754 Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓)
 
Theoremexanali 1755 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))
 
Theoremexancom 1756 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))
 
Theoremexan 1757 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.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)
 
Theoremalrimdh 1758 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2036 and 19.21h 2038. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))
 
Theoremeximdh 1759 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theoremnexdh 1760 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)
 
Theoremalbidh 1761 Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))
 
Theoremexbidh 1762 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
TheoremexbidhOLD 1763 Obsolete proof of exbidh 1762 as of 16-Nov-2020. (Contributed by NM, 26-May-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))
 
Theoremexsimpl 1764 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 1765 Simplification of an existentially quantified conjunction. (Contributed by Rodolfo Medina, 25-Sep-2010.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
(∃𝑥(𝜑𝜓) → ∃𝑥𝜓)
 
Theorem19.40 1766 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
(∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))
 
Theorem19.26 1767 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 1768 Theorem 19.26 1767 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓))
 
Theorem19.26-3an 1769 Theorem 19.26 1767 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
(∀𝑥(𝜑𝜓𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))
 
Theorem19.29 1770 Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1771. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.29r 1771 Variation of 19.29 1770. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.29rOLD 1772 Obsolete proof of 19.29r 1771 as 12-Nov-2020. (Contributed by NM, 18-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))
 
Theorem19.29r2 1773 Variation of 19.29r 1771 with double quantification. (Contributed by NM, 3-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
 
Theorem19.29x 1774 Variation of 19.29 1770 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))
 
Theorem19.35 1775 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 1776 Inference associated with 19.35 1775. (Contributed by NM, 21-Jun-1993.)
𝑥(𝜑𝜓)       (∀𝑥𝜑 → ∃𝑥𝜓)
 
Theorem19.35ri 1777 Inference associated with 19.35 1775. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝜑 → ∃𝑥𝜓)       𝑥(𝜑𝜓)
 
Theorem19.25 1778 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑦𝑥(𝜑𝜓) → (∃𝑦𝑥𝜑 → ∃𝑦𝑥𝜓))
 
Theorem19.30 1779 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 1780 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 1781 Obsolete proof of 19.43 1780. Do not delete as it is referenced on the mmrecent.html page and in conventions-label 26389. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))
 
Theorem19.33 1782 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))
 
Theorem19.33b 1783 The antecedent provides a condition implying the converse of 19.33 1782. (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 1784 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 1785 The antecedent provides a condition implying the converse of 19.40 1766. This is to 19.40 1766 what 19.33b 1783 is to 19.33 1782. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))
 
Theorem19.40bOLD 1786 Obsolete proof of 19.40b 1785 as of 13-Nov-2020. (Contributed by BJ, 6-May-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))
 
Theoremalbiim 1787 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))
 
Theorem2albiim 1788 Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))
 
Theoremexintrbi 1789 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))
 
TheoremexintrbiOLD 1790 Obsolete proof of exintrbi 1789 as of 16-Nov-2020. (Contributed by Raph Levien, 3-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))
 
Theoremexintr 1791 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))
 
Theoremalsyl 1792 Theorem *10.3 in [WhiteheadRussell] p. 150. (Contributed by Andrew Salmon, 8-Jun-2011.)
((∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜒)) → ∀𝑥(𝜑𝜒))
 
1.4.4  Axiom scheme ax-5 (Distinctness) - first use of $d
 
Axiomax-5 1793* 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 33100 about the logical redundancy of ax-5 1793 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 1990, we can also remove the quantifier (unconditionally). (Contributed by NM, 10-Jan-1993.)

(𝜑 → ∀𝑥𝜑)
 
Theoremax5d 1794* ax-5 1793 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
(𝜑 → (𝜓 → ∀𝑥𝜓))
 
Theoremax5e 1795* A rephrasing of ax-5 1793 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)
 
Theoremnfv 1796* If 𝑥 is not present in 𝜑, then 𝑥 is not free in 𝜑. (Contributed by Mario Carneiro, 11-Aug-2016.)
𝑥𝜑
 
Theoremnfvd 1797* nfv 1796 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 2048. (Contributed by Mario Carneiro, 6-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)
 
Theoremalimdv 1798* Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1714. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))
 
Theoremeximdv 1799* Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1739. (Contributed by NM, 27-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))
 
Theorem2alimdv 1800* Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1714. (Contributed by NM, 27-Apr-2004.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))
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