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Theorem List for Metamath Proof Explorer - 1901-2000   *Has distinct variable group(s)
TypeLabelDescription
Statement

Theorem2exnaln 1901 Theorem *11.22 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(∃𝑥𝑦𝜑 ↔ ¬ ∀𝑥𝑦 ¬ 𝜑)

Theorem2nexaln 1902 Theorem *11.25 in [WhiteheadRussell] p. 160. (Contributed by Andrew Salmon, 24-May-2011.)
(¬ ∃𝑥𝑦𝜑 ↔ ∀𝑥𝑦 ¬ 𝜑)

Theoremalimex 1903 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 1852. (Contributed by BJ, 12-May-2019.)
((𝜑 → ∀𝑥𝜓) ↔ (∃𝑥 ¬ 𝜓 → ¬ 𝜑))

Theoremaleximi 1904 A variant of al2imi 1888: instead of applying 𝑥 quantifiers to the final implication, replace them with 𝑥. A shorter proof is possible using nfa1 2173, sps 2198 and eximd 2228, but it depends on more axioms. (Contributed by Wolf Lammen, 18-Aug-2019.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremalexbii 1905 Biconditional form of aleximi 1904. (Contributed by BJ, 16-Nov-2020.)
(𝜑 → (𝜓𝜒))       (∀𝑥𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremexim 1906 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 1907 Inference adding existential quantifier to antecedent and consequent. (Contributed by NM, 10-Jan-1993.)
(𝜑𝜓)       (∃𝑥𝜑 → ∃𝑥𝜓)

Theorem2eximi 1908 Inference adding two existential quantifiers to antecedent and consequent. (Contributed by NM, 3-Feb-2005.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 → ∃𝑥𝑦𝜓)

Theoremeximii 1909 Inference associated with eximi 1907. (Contributed by BJ, 3-Feb-2018.)
𝑥𝜑    &   (𝜑𝜓)       𝑥𝜓

Theoremexa1 1910 Add an antecedent in an existentially quantified formula. (Contributed by BJ, 6-Oct-2018.)
(∃𝑥𝜑 → ∃𝑥(𝜓𝜑))

Theorem19.38 1911 Theorem 19.38 of [Margaris] p. 90. The converse holds under non-freeness conditions, see 19.38a 1912 and 19.38b 1913. (Contributed by NM, 12-Mar-1993.) Allow a shortening of 19.21t 2216. (Revised by Wolf Lammen, 2-Jan-2018.)
((∃𝑥𝜑 → ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.38a 1912 Under a non-freeness hypothesis, the implication 19.38 1911 can be strengthened to an equivalence. See also 19.38b 1913. (Contributed by BJ, 3-Nov-2021.)
(Ⅎ𝑥𝜑 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Theorem19.38b 1913 Under a non-freeness hypothesis, the implication 19.38 1911 can be strengthened to an equivalence. See also 19.38a 1912. (Contributed by BJ, 3-Nov-2021.)
(Ⅎ𝑥𝜓 → ((∃𝑥𝜑 → ∀𝑥𝜓) ↔ ∀𝑥(𝜑𝜓)))

Theoremimnang 1914 Quantified implication in terms of quantified negation of conjunction. (Contributed by BJ, 16-Jul-2021.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ∀𝑥 ¬ (𝜑𝜓))

Theoremalinexa 1915 A transformation of quantifiers and logical connectives. (Contributed by NM, 19-Aug-1993.)
(∀𝑥(𝜑 → ¬ 𝜓) ↔ ¬ ∃𝑥(𝜑𝜓))

Theoremalexn 1916 A relationship between two quantifiers and negation. (Contributed by NM, 18-Aug-1993.)
(∀𝑥𝑦 ¬ 𝜑 ↔ ¬ ∃𝑥𝑦𝜑)

Theorem2exnexn 1917 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 1918 Theorem 19.18 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥𝜓))

Theoremexbii 1919 Inference adding existential quantifier to both sides of an equivalence. (Contributed by NM, 24-May-1994.)
(𝜑𝜓)       (∃𝑥𝜑 ↔ ∃𝑥𝜓)

Theorem2exbii 1920 Inference adding two existential quantifiers to both sides of an equivalence. (Contributed by NM, 16-Mar-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝜑 ↔ ∃𝑥𝑦𝜓)

Theorem3exbii 1921 Inference adding three existential quantifiers to both sides of an equivalence. (Contributed by NM, 2-May-1995.)
(𝜑𝜓)       (∃𝑥𝑦𝑧𝜑 ↔ ∃𝑥𝑦𝑧𝜓)

Theoremnfbiit 1922 Equivalence theorem for the non-freeness predicate. Closed form of nfbii 1923. (Contributed by BJ, 6-May-2019.)
(∀𝑥(𝜑𝜓) → (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓))

Theoremnfbii 1923 Equality theorem for the non-freeness predicate. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1855 changed. (Revised by Wolf Lammen, 12-Sep-2021.)
(𝜑𝜓)       (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

Theoremnfxfr 1924 A utility lemma to transfer a bound-variable hypothesis builder into a definition. (Contributed by Mario Carneiro, 11-Aug-2016.)
(𝜑𝜓)    &   𝑥𝜓       𝑥𝜑

Theoremnfxfrd 1925 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 1926 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 1927 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 1855 changed. (Revised by Wolf Lammen, 4-Oct-2021.)
(Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

TheoremnfntOLDOLD 1928 Obsolete proof of nfnt 1927 as of 3-Nov-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 28-Dec-2017.) (Revised by BJ, 24-Jul-2019.) df-nf 1855 changed. (Revised by Wolf Lammen, 4-Oct-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(Ⅎ𝑥𝜑 → Ⅎ𝑥 ¬ 𝜑)

Theoremnfn 1929 Inference associated with nfnt 1927. (Contributed by Mario Carneiro, 11-Aug-2016.) df-nf 1855 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
𝑥𝜑       𝑥 ¬ 𝜑

Theoremnfnd 1930 Deduction associated with nfnt 1927. (Contributed by Mario Carneiro, 24-Sep-2016.)
(𝜑 → Ⅎ𝑥𝜓)       (𝜑 → Ⅎ𝑥 ¬ 𝜓)

Theoremexanali 1931 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 1932 Commutation of conjunction inside an existential quantifier. (Contributed by NM, 18-Aug-1993.)
(∃𝑥(𝜑𝜓) ↔ ∃𝑥(𝜓𝜑))

Theoremexan 1933 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, 8-Oct-2021.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)

TheoremexanOLD 1934 Obsolete proof of exan 1933 as of 8-Oct-2021. (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 modification is discouraged.) (New usage is discouraged.)
(∃𝑥𝜑𝜓)       𝑥(𝜑𝜓)

Theoremalrimdh 1935 Deduction form of Theorem 19.21 of [Margaris] p. 90, see 19.21 2218 and 19.21h 2264. (Contributed by NM, 10-Feb-1997.) (Proof shortened by Andrew Salmon, 13-May-2011.)
(𝜑 → ∀𝑥𝜑)    &   (𝜓 → ∀𝑥𝜓)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (𝜓 → ∀𝑥𝜒))

Theoremeximdh 1936 Deduction from Theorem 19.22 of [Margaris] p. 90. (Contributed by NM, 20-May-1996.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theoremnexdh 1937 Deduction for generalization rule for negated wff. (Contributed by NM, 2-Jan-2002.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → ¬ 𝜓)       (𝜑 → ¬ ∃𝑥𝜓)

Theoremalbidh 1938 Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbidh 1939 Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → ∀𝑥𝜑)    &   (𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theoremexsimpl 1940 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 1941 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 1942 Theorem 19.40 of [Margaris] p. 90. (Contributed by NM, 26-May-1993.)
(∃𝑥(𝜑𝜓) → (∃𝑥𝜑 ∧ ∃𝑥𝜓))

Theorem19.26 1943 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 1944 Theorem 19.26 1943 with two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓))

Theorem19.26-3an 1945 Theorem 19.26 1943 with triple conjunction. (Contributed by NM, 13-Sep-2011.)
(∀𝑥(𝜑𝜓𝜒) ↔ (∀𝑥𝜑 ∧ ∀𝑥𝜓 ∧ ∀𝑥𝜒))

Theorem19.29 1946 Theorem 19.29 of [Margaris] p. 90. See also 19.29r 1947. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Andrew Salmon, 13-May-2011.)
((∀𝑥𝜑 ∧ ∃𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.29r 1947 Variation of 19.29 1946. (Contributed by NM, 18-Aug-1993.) (Proof shortened by Wolf Lammen, 12-Nov-2020.)
((∃𝑥𝜑 ∧ ∀𝑥𝜓) → ∃𝑥(𝜑𝜓))

Theorem19.29r2 1948 Variation of 19.29r 1947 with double quantification. (Contributed by NM, 3-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

Theorem19.29x 1949 Variation of 19.29 1946 with mixed quantification. (Contributed by NM, 11-Feb-2005.)
((∃𝑥𝑦𝜑 ∧ ∀𝑥𝑦𝜓) → ∃𝑥𝑦(𝜑𝜓))

Theorem19.35 1950 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 1951 Inference associated with 19.35 1950. (Contributed by NM, 21-Jun-1993.)
𝑥(𝜑𝜓)       (∀𝑥𝜑 → ∃𝑥𝜓)

Theorem19.35ri 1952 Inference associated with 19.35 1950. (Contributed by NM, 12-Mar-1993.)
(∀𝑥𝜑 → ∃𝑥𝜓)       𝑥(𝜑𝜓)

Theorem19.25 1953 Theorem 19.25 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
(∀𝑦𝑥(𝜑𝜓) → (∃𝑦𝑥𝜑 → ∃𝑦𝑥𝜓))

Theorem19.30 1954 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 1955 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 1956 Obsolete proof of 19.43 1955. Do not delete as it is referenced on the mmrecent.html page and in conventions-label 27565. (Contributed by NM, 5-Aug-1993.) (Proof modification is discouraged.) (New usage is discouraged.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑 ∨ ∃𝑥𝜓))

Theorem19.33 1957 Theorem 19.33 of [Margaris] p. 90. (Contributed by NM, 12-Mar-1993.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ∀𝑥(𝜑𝜓))

Theorem19.33b 1958 The antecedent provides a condition implying the converse of 19.33 1957. (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 1959 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 1960 The antecedent provides a condition implying the converse of 19.40 1942. This is to 19.40 1942 what 19.33b 1958 is to 19.33 1957. (Contributed by BJ, 6-May-2019.) (Proof shortened by Wolf Lammen, 13-Nov-2020.)
((∀𝑥𝜑 ∨ ∀𝑥𝜓) → ((∃𝑥𝜑 ∧ ∃𝑥𝜓) ↔ ∃𝑥(𝜑𝜓)))

Theoremalbiim 1961 Split a biconditional and distribute quantifier. (Contributed by NM, 18-Aug-1993.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥(𝜑𝜓) ∧ ∀𝑥(𝜓𝜑)))

Theorem2albiim 1962 Split a biconditional and distribute two quantifiers. (Contributed by NM, 3-Feb-2005.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∀𝑥𝑦(𝜑𝜓) ∧ ∀𝑥𝑦(𝜓𝜑)))

Theoremexintrbi 1963 Add/remove a conjunct in the scope of an existential quantifier. (Contributed by Raph Levien, 3-Jul-2006.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 ↔ ∃𝑥(𝜑𝜓)))

Theoremexintr 1964 Introduce a conjunct in the scope of an existential quantifier. (Contributed by NM, 11-Aug-1993.)
(∀𝑥(𝜑𝜓) → (∃𝑥𝜑 → ∃𝑥(𝜑𝜓)))

Theoremalsyl 1965 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 1966 Closed form of nfim 1970 and curried (exported) form of nfimt2 1967. (Contributed by BJ, 20-Oct-2021.)
(Ⅎ𝑥𝜑 → (Ⅎ𝑥𝜓 → Ⅎ𝑥(𝜑𝜓)))

Theoremnfimt2 1967 Closed form of nfim 1970 and uncurried (imported) form of nfimt 1966. (Contributed by BJ, 20-Oct-2021.)
((Ⅎ𝑥𝜑 ∧ Ⅎ𝑥𝜓) → Ⅎ𝑥(𝜑𝜓))

Theoremnfimd 1968 If in a context 𝑥 is not free in 𝜓 and 𝜒, it is not free in (𝜓𝜒). Deduction form of nfimt 1966. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1855 changed. (Revised by Wolf Lammen, 18-Sep-2021.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

TheoremnfimdOLDOLD 1969 Obsolete proof of nfimd 1968 as of 3-Nov-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2017.) df-nf 1855 changed. (Revised by Wolf Lammen, 18-Sep-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)       (𝜑 → Ⅎ𝑥(𝜓𝜒))

Theoremnfim 1970 If 𝑥 is not free in 𝜑 and 𝜓, it is not free in (𝜑𝜓). Inference associated with nfimt 1966. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 2-Jan-2018.) df-nf 1855 changed. (Revised by Wolf Lammen, 17-Sep-2021.)
𝑥𝜑    &   𝑥𝜓       𝑥(𝜑𝜓)

Theoremnfand 1971 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 1972 Deduction form of bound-variable hypothesis builder nf3an 1976. (Contributed by NM, 17-Feb-2013.) (Revised by Mario Carneiro, 16-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)    &   (𝜑 → Ⅎ𝑥𝜒)    &   (𝜑 → Ⅎ𝑥𝜃)       (𝜑 → Ⅎ𝑥(𝜓𝜒𝜃))

Theoremnfan 1973 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 1974 Obsolete proof of nfan 1973 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 1975 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 1976 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 1977 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 1978 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 1979 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 1980 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 1981 Obsolete proof of nfbii 1923 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)       (Ⅎ𝑥𝜑 ↔ Ⅎ𝑥𝜓)

TheoremnfxfrOLD 1982 Obsolete proof of nfxfr 1924 as of 6-Oct-2021. (Contributed by Mario Carneiro, 11-Aug-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝜑𝜓)    &   𝑥𝜓       𝑥𝜑

TheoremnfxfrdOLD 1983 Obsolete proof of nfxfrd 1925 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 1984* 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 34692 about the logical redundancy of ax-5 1984 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 2196, we can also remove the quantifier (unconditionally). (Contributed by NM, 10-Jan-1993.)

(𝜑 → ∀𝑥𝜑)

Theoremax5d 1985* ax-5 1984 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders. (Contributed by NM, 1-Mar-2013.)
(𝜑 → (𝜓 → ∀𝑥𝜓))

Theoremax5e 1986* A rephrasing of ax-5 1984 using the existential quantifier. (Contributed by Wolf Lammen, 4-Dec-2017.)
(∃𝑥𝜑𝜑)

Theoremax5ea 1987* 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 1988* 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 1989* nfv 1988 with antecedent. Useful in proofs of deduction versions of bound-variable hypothesis builders such as nfimd 1968. (Contributed by Mario Carneiro, 6-Oct-2016.)
(𝜑 → Ⅎ𝑥𝜓)

Theoremalimdv 1990* Deduction form of Theorem 19.20 of [Margaris] p. 90, see alim 1883. (Contributed by NM, 3-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 → ∀𝑥𝜒))

Theoremeximdv 1991* Deduction form of Theorem 19.22 of [Margaris] p. 90, see exim 1906. (Contributed by NM, 27-Apr-1994.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 → ∃𝑥𝜒))

Theorem2alimdv 1992* Deduction form of Theorem 19.20 of [Margaris] p. 90 with two quantifiers, see alim 1883. (Contributed by NM, 27-Apr-2004.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 → ∀𝑥𝑦𝜒))

Theorem2eximdv 1993* Deduction form of Theorem 19.22 of [Margaris] p. 90 with two quantifiers, see exim 1906. (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 → ∃𝑥𝑦𝜒))

Theoremalbidv 1994* Formula-building rule for universal quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝜓 ↔ ∀𝑥𝜒))

Theoremexbidv 1995* Formula-building rule for existential quantifier (deduction rule). (Contributed by NM, 26-May-1993.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝜓 ↔ ∃𝑥𝜒))

Theorem2albidv 1996* Formula-building rule for two universal quantifiers (deduction rule). (Contributed by NM, 4-Mar-1997.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∀𝑥𝑦𝜓 ↔ ∀𝑥𝑦𝜒))

Theorem2exbidv 1997* Formula-building rule for two existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝜓 ↔ ∃𝑥𝑦𝜒))

Theorem3exbidv 1998* Formula-building rule for three existential quantifiers (deduction rule). (Contributed by NM, 1-May-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝜓 ↔ ∃𝑥𝑦𝑧𝜒))

Theorem4exbidv 1999* Formula-building rule for four existential quantifiers (deduction rule). (Contributed by NM, 3-Aug-1995.)
(𝜑 → (𝜓𝜒))       (𝜑 → (∃𝑥𝑦𝑧𝑤𝜓 ↔ ∃𝑥𝑦𝑧𝑤𝜒))

Theoremalrimiv 2000* Inference form of Theorem 19.21 of [Margaris] p. 90. See 19.21 2218 and 19.21v 2013. (Contributed by NM, 21-Jun-1993.)
(𝜑𝜓)       (𝜑 → ∀𝑥𝜓)

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