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

Theorem19.23vv 1901* Theorem 19.23v 1900 extended to two variables. (Contributed by NM, 10-Aug-2004.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theorem19.36v 1902* Version of 19.36 2096 with a dv condition instead of a non-freeness hypothesis. (Contributed by NM, 18-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
(∃𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.36iv 1903* Inference associated with 19.36v 1902. Version of 19.36i 2097 with a dv condition. (Contributed by NM, 5-Aug-1993.) Reduce dependencies on axioms. (Revised by Wolf Lammen, 17-Jan-2020.)
𝑥(𝜑𝜓)       (∀𝑥𝜑𝜓)

Theorempm11.53v 1904* Version of pm11.53 2177 with a dv condition, requiring fewer axioms. (Contributed by BJ, 7-Mar-2020.)
(∀𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝜑 → ∀𝑦𝜓))

Theorem19.12vvv 1905* Version of 19.12vv 2178 with a dv condition, requiring fewer axioms. See also 19.12 2162. (Contributed by BJ, 18-Mar-2020.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∀𝑦𝑥(𝜑𝜓))

Theorem19.27v 1906* Version of 19.27 2093 with a dv condition, requiring fewer axioms. (Contributed by NM, 3-Jun-2004.)
(∀𝑥(𝜑𝜓) ↔ (∀𝑥𝜑𝜓))

Theorem19.28v 1907* Version of 19.28 2094 with a dv condition, requiring fewer axioms. (Contributed by NM, 25-Mar-2004.)
(∀𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))

Theorem19.37v 1908* Version of 19.37 2098 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 → ∃𝑥𝜓))

Theorem19.37iv 1909* Inference associated with 19.37v 1908. (Contributed by NM, 5-Aug-1993.)
𝑥(𝜑𝜓)       (𝜑 → ∃𝑥𝜓)

Theorem19.44v 1910* Version of 19.44 2104 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.45v 1911* Version of 19.45 2105 with a dv condition, requiring fewer axioms. (Contributed by NM, 12-Mar-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∨ ∃𝑥𝜓))

Theorem19.41v 1912* Version of 19.41 2101 with a dv condition, requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (∃𝑥𝜑𝜓))

Theorem19.41vv 1913* Version of 19.41 2101 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (∃𝑥𝑦𝜑𝜓))

Theorem19.41vvv 1914* Version of 19.41 2101 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 30-Apr-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑥𝑦𝑧𝜑𝜓))

Theorem19.41vvvv 1915* Version of 19.41 2101 with four quantifiers and a dv condition requiring fewer axioms. (Contributed by FL, 14-Jul-2007.)
(∃𝑤𝑥𝑦𝑧(𝜑𝜓) ↔ (∃𝑤𝑥𝑦𝑧𝜑𝜓))

Theorem19.42v 1916* Version of 19.42 2103 with a dv condition requiring fewer axioms. (Contributed by NM, 21-Jun-1993.)
(∃𝑥(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝜓))

Theoremexdistr 1917* Distribution of existential quantifiers. (Contributed by NM, 9-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝜓))

Theorem19.42vv 1918* Version of 19.42 2103 with two quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 16-Mar-1995.)
(∃𝑥𝑦(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝜓))

Theorem19.42vvv 1919* Version of 19.42 2103 with three quantifiers and a dv condition requiring fewer axioms. (Contributed by NM, 21-Sep-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ (𝜑 ∧ ∃𝑥𝑦𝑧𝜓))

Theoremexdistr2 1920* Distribution of existential quantifiers. (Contributed by NM, 17-Mar-1995.)
(∃𝑥𝑦𝑧(𝜑𝜓) ↔ ∃𝑥(𝜑 ∧ ∃𝑦𝑧𝜓))

Theorem3exdistr 1921* Distribution of existential quantifiers in a triple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Andrew Salmon, 25-May-2011.)
(∃𝑥𝑦𝑧(𝜑𝜓𝜒) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧𝜒)))

Theorem4exdistr 1922* Distribution of existential quantifiers in a quadruple conjunction. (Contributed by NM, 9-Mar-1995.) (Proof shortened by Wolf Lammen, 20-Jan-2018.)
(∃𝑥𝑦𝑧𝑤((𝜑𝜓) ∧ (𝜒𝜃)) ↔ ∃𝑥(𝜑 ∧ ∃𝑦(𝜓 ∧ ∃𝑧(𝜒 ∧ ∃𝑤𝜃))))

Theoremspimeh 1923* Existential introduction, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
(𝜑 → ∀𝑥𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (𝜑 → ∃𝑥𝜓)

Theoremspimw 1924* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 7-Aug-2017.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspimvw 1925* Specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜓)

Theoremspnfw 1926 Weak version of sp 2051. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 1-Aug-2017.) (Proof shortened by Wolf Lammen, 13-Aug-2017.)
𝜑 → ∀𝑥 ¬ 𝜑)       (∀𝑥𝜑𝜑)

Theoremspfalw 1927 Version of sp 2051 when 𝜑 is false. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 23-Apr-2017.) (Proof shortened by Wolf Lammen, 25-Dec-2017.)
¬ 𝜑       (∀𝑥𝜑𝜑)

Theoremequs4v 1928* Version of equs4 2288 with a dv condition, which requires fewer axioms. (Contributed by BJ, 31-May-2019.)
(∀𝑥(𝑥 = 𝑦𝜑) → ∃𝑥(𝑥 = 𝑦𝜑))

Theoremequsalvw 1929* Version of equsalv 2106 with a dv condition, and of equsal 2289 with two dv conditions, which requires fewer axioms. See also the dual form equsexvw 1930. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremequsexvw 1930* Version of equsexv 2107 with a dv condition, and of equsex 2290 with two dv conditions, which requires fewer axioms. See also the dual form equsalvw 1929. (Contributed by BJ, 31-May-2019.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥(𝑥 = 𝑦𝜑) ↔ 𝜓)

Theoremcbvaliw 1931* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 19-Apr-2017.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

Theoremcbvalivw 1932* Change bound variable. Uses only Tarski's FOL axiom schemes. Part of Lemma 7 of [KalishMontague] p. 86. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑦𝜓)

1.4.8  Axiom scheme ax-7 (Equality)

Axiomax-7 1933 Axiom of Equality. One of the equality and substitution axioms of predicate calculus with equality. It states that equality is a right-Euclidean binary relation (this is similar, but not identical, to being transitive, which is proved as equtr 1946). This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom C7 of [Monk2] p. 105 and Axiom Scheme C8' in [Megill] p. 448 (p. 16 of the preprint).

The equality symbol was invented in 1557 by Robert Recorde. He chose a pair of parallel lines of the same length because "noe .2. thynges, can be moare equalle."

We prove in ax7 1941 that this axiom can be recovered from its weakened version ax7v 1934 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-7 1933 should be ax7v 1934. See the comment of ax7v 1934 for more details on these matters. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 7-Dec-2020.) Use ax7 1941 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v 1934* Weakened version of ax-7 1933, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-7 1933, and it should be referenced only by its two weakened versions ax7v1 1935 and ax7v2 1936, from which ax-7 1933 is then rederived as ax7 1941, which shows that either ax7v 1934 or the conjunction of ax7v1 1935 and ax7v2 1936 is sufficient.

In ax7v 1934, it is still allowed to substitute the same variable for 𝑥 and 𝑧, or the same variable for 𝑦 and 𝑧. Therefore, ax7v 1934 "bundles" (a term coined by Raph Levien) its "principal instance" (𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧)) with 𝑥, 𝑦, 𝑧 distinct, and its "degenerate instances" (𝑥 = 𝑦 → (𝑥 = 𝑥𝑦 = 𝑥)) and (𝑥 = 𝑦 → (𝑥 = 𝑦𝑦 = 𝑦)) with 𝑥, 𝑦 distinct. These degenerate instances are for instance used in the proofs of equcomiv 1939 and equid 1937 respectively. (Contributed by BJ, 7-Dec-2020.) Use ax7 1941 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v1 1935* First of two weakened versions of ax7v 1934, with an extra dv condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremax7v2 1936* Second of two weakened versions of ax7v 1934, with an extra dv condition on 𝑦, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequid 1937 Identity law for equality. Lemma 2 of [KalishMontague] p. 85. See also Lemma 6 of [Tarski] p. 68. (Contributed by NM, 1-Apr-2005.) (Revised by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 5-Feb-2018.) (Proof shortened by Wolf Lammen, 22-Aug-2020.)
𝑥 = 𝑥

Theoremnfequid 1938 Bound-variable hypothesis builder for 𝑥 = 𝑥. This theorem tells us that any variable, including 𝑥, is effectively not free in 𝑥 = 𝑥, even though 𝑥 is technically free according to the traditional definition of free variable. (Contributed by NM, 13-Jan-2011.) (Revised by NM, 21-Aug-2017.)
𝑦 𝑥 = 𝑥

Theoremequcomiv 1939* Weaker form of equcomi 1942 with a dv condition on 𝑥, 𝑦. This is an intermediate step and equcomi 1942 is fully recovered later. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremax6evr 1940* A commuted form of ax6ev 1888. (Contributed by BJ, 7-Dec-2020.)
𝑥 𝑦 = 𝑥

Theoremax7 1941 Proof of ax-7 1933 from ax7v1 1935 and ax7v2 1936, proving sufficiency of the conjunction of the latter two weakened versions of ax7v 1934, which is itself a weakened version of ax-7 1933.

Note that the weakened version of ax-7 1933 obtained by adding a dv condition on 𝑥, 𝑧 (resp. on 𝑦, 𝑧) does not permit, together with the other axioms, to prove reflexivity (resp. symmetry). (Contributed by BJ, 7-Dec-2020.)

(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequcomi 1942 Commutative law for equality. Equality is a symmetric relation. Lemma 3 of [KalishMontague] p. 85. See also Lemma 7 of [Tarski] p. 69. (Contributed by NM, 10-Jan-1993.) (Revised by NM, 9-Apr-2017.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcom 1943 Commutative law for equality. Equality is a symmetric relation. (Contributed by NM, 20-Aug-1993.)
(𝑥 = 𝑦𝑦 = 𝑥)

Theoremequcomd 1944 Deduction form of equcom 1943, symmetry of equality. For the versions for classes, see eqcom 2627 and eqcomd 2626. (Contributed by BJ, 6-Oct-2019.)
(𝜑𝑥 = 𝑦)       (𝜑𝑦 = 𝑥)

Theoremequcoms 1945 An inference commuting equality in antecedent. Used to eliminate the need for a syllogism. (Contributed by NM, 10-Jan-1993.)
(𝑥 = 𝑦𝜑)       (𝑦 = 𝑥𝜑)

Theoremequtr 1946 A transitive law for equality. (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑦 = 𝑧𝑥 = 𝑧))

Theoremequtrr 1947 A transitive law for equality. Lemma L17 in [Megill] p. 446 (p. 14 of the preprint). (Contributed by NM, 23-Aug-1993.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Theoremequeuclr 1948 Commuted version of equeucl 1949 (equality is left-Euclidean). (Contributed by BJ, 12-Apr-2021.)
(𝑥 = 𝑧 → (𝑦 = 𝑧𝑦 = 𝑥))

Theoremequeucl 1949 Equality is a left-Euclidean binary relation. (Right-Euclideanness is stated in ax-7 1933.) Curried (exported) form of equtr2 1952. (Contributed by BJ, 11-Apr-2021.)
(𝑥 = 𝑧 → (𝑦 = 𝑧𝑥 = 𝑦))

Theoremequequ1 1950 An equivalence law for equality. (Contributed by NM, 1-Aug-1993.) (Proof shortened by Wolf Lammen, 10-Dec-2017.)
(𝑥 = 𝑦 → (𝑥 = 𝑧𝑦 = 𝑧))

Theoremequequ2 1951 An equivalence law for equality. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) (Proof shortened by BJ, 12-Apr-2021.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Theoremequtr2 1952 Equality is a left-Euclidean binary relation. Uncurried (imported) form of equeucl 1949. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof shortened by BJ, 11-Apr-2021.)
((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)

Theoremequequ2OLD 1953 Obsolete proof of equequ2 1951 as of 12-Apr-2021. (Contributed by NM, 21-Jun-1993.) (Proof shortened by Wolf Lammen, 4-Aug-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑧 = 𝑥𝑧 = 𝑦))

Theoremequtr2OLD 1954 Obsolete proof of equtr2 1952 as of 11-Apr-2021. (Contributed by NM, 12-Aug-1993.) (Proof shortened by Andrew Salmon, 25-May-2011.) (Proof modification is discouraged.) (New usage is discouraged.)
((𝑥 = 𝑧𝑦 = 𝑧) → 𝑥 = 𝑦)

Theoremstdpc6 1955 One of the two equality axioms of standard predicate calculus, called reflexivity of equality. (The other one is stdpc7 1956.) Axiom 6 of [Mendelson] p. 95. Mendelson doesn't say why he prepended the redundant quantifier, but it was probably to be compatible with free logic (which is valid in the empty domain). (Contributed by NM, 16-Feb-2005.)
𝑥 𝑥 = 𝑥

Theoremstdpc7 1956 One of the two equality axioms of standard predicate calculus, called substitutivity of equality. (The other one is stdpc6 1955.) Translated to traditional notation, it can be read: "𝑥 = 𝑦 → (𝜑(𝑥, 𝑥) → 𝜑(𝑥, 𝑦)), provided that 𝑦 is free for 𝑥 in 𝜑(𝑥, 𝑥)." Axiom 7 of [Mendelson] p. 95. (Contributed by NM, 15-Feb-2005.)
(𝑥 = 𝑦 → ([𝑥 / 𝑦]𝜑𝜑))

Theoremequvinv 1957* A variable introduction law for equality. Lemma 15 of [Monk2] p. 109. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2017, ax-13 2244. (Revised by Wolf Lammen, 10-Jun-2019.) Move the quantified variable (𝑧) to the left of the equality signs. (Revised by Wolf Lammen, 11-Apr-2021.)
(𝑥 = 𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))

Theoremequviniva 1958* A modified version of the forward implication of equvinv 1957 adapted to common usage. (Contributed by Wolf Lammen, 8-Sep-2018.)
(𝑥 = 𝑦 → ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))

TheoremequvinivOLD 1959* The forward implication of equvinv 1957. Obsolete as of 11-Apr-2021. Use equvinv 1957 instead. (Contributed by Wolf Lammen, 11-Apr-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → ∃𝑧(𝑧 = 𝑥𝑧 = 𝑦))

TheoremequvinvOLD 1960* Obsolete version of equvinv 1957 as of 11-Apr-2021. (Contributed by NM, 9-Jan-1993.) Remove dependencies on ax-10 2017, ax-13 2244. (Revised by Wolf Lammen, 10-Jun-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 ↔ ∃𝑧(𝑥 = 𝑧𝑦 = 𝑧))

Theoremequvelv 1961* A specialized version of equvel 2345 with distinct variable restrictions and fewer axiom usage. (Contributed by Wolf Lammen, 10-Apr-2021.)
(𝑥 = 𝑦 ↔ ∀𝑧(𝑧 = 𝑥𝑧 = 𝑦))

Theoremax13b 1962 An equivalence between two ways of expressing ax-13 2244. See the comment for ax-13 2244. (Contributed by NM, 2-May-2017.) (Proof shortened by Wolf Lammen, 26-Feb-2018.) (Revised by BJ, 15-Sep-2020.)
((¬ 𝑥 = 𝑦 → (𝑦 = 𝑧𝜑)) ↔ (¬ 𝑥 = 𝑦 → (¬ 𝑥 = 𝑧 → (𝑦 = 𝑧𝜑))))

Theoremspfw 1963* Weak version of sp 2051. Uses only Tarski's FOL axiom schemes. Lemma 9 of [KalishMontague] p. 87. This may be the best we can do with minimal distinct variable conditions. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)

TheoremspfwOLD 1964* Obsolete proof of spfw 1963 as of 10-Oct-2021. (Contributed by NM, 19-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)

Theoremspw 1965* Weak version of the specialization scheme sp 2051. Lemma 9 of [KalishMontague] p. 87. While it appears that sp 2051 in its general form does not follow from Tarski's FOL axiom schemes, from this theorem we can prove any instance of sp 2051 having mutually distinct setvar variables and no wff metavariables (see ax12wdemo 2010 for an example of the procedure to eliminate the hypothesis). Other approximations of sp 2051 are spfw 1963 (minimal distinct variable requirements), spnfw 1926 (when 𝑥 is not free in ¬ 𝜑), spvw 1896 (when 𝑥 does not appear in 𝜑), sptruw 1731 (when 𝜑 is true), and spfalw 1927 (when 𝜑 is false). (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 27-Feb-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑𝜑)

Theoremcbvalw 1966* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑦𝜓 → ∀𝑥𝑦𝜓)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvalvw 1967* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Theoremcbvexvw 1968* Change bound variable. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Theoremalcomiw 1969* Weak version of alcom 2035. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 10-Apr-2017.)
(𝑦 = 𝑧 → (𝜑𝜓))       (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Theoremhbn1fw 1970* Weak version of ax-10 2017 from which we can prove any ax-10 2017 instance not involving wff variables or bundling. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.) (Proof shortened by Wolf Lammen, 28-Feb-2018.)
(∀𝑥𝜑 → ∀𝑦𝑥𝜑)    &   𝜓 → ∀𝑥 ¬ 𝜓)    &   (∀𝑦𝜓 → ∀𝑥𝑦𝜓)    &   𝜑 → ∀𝑦 ¬ 𝜑)    &   (¬ ∀𝑦𝜓 → ∀𝑥 ¬ ∀𝑦𝜓)    &   (𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhbn1w 1971* Weak version of hbn1 2018. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (¬ ∀𝑥𝜑 → ∀𝑥 ¬ ∀𝑥𝜑)

Theoremhba1w 1972* Weak version of hba1 2149. See comments for ax10w 2004. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 9-Apr-2017.) (Proof shortened by Wolf Lammen, 10-Oct-2021.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhba1wOLD 1973* Obsolete proof of hba1w 1972 as of 10-Oct-2021. (Contributed by NM, 9-Apr-2017.) (Proof modification is discouraged.) (New usage is discouraged.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∀𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhbe1w 1974* Weak version of hbe1 2019. See comments for ax10w 2004. Uses only Tarski's FOL axiom schemes. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑦 → (𝜑𝜓))       (∃𝑥𝜑 → ∀𝑥𝑥𝜑)

Theoremhbalw 1975* Weak version of hbal 2034. Uses only Tarski's FOL axiom schemes. Unlike hbal 2034, this theorem requires that 𝑥 and 𝑦 be distinct, i.e. not be bundled. (Contributed by NM, 19-Apr-2017.)
(𝑥 = 𝑧 → (𝜑𝜓))    &   (𝜑 → ∀𝑥𝜑)       (∀𝑦𝜑 → ∀𝑥𝑦𝜑)

Theoremspaev 1976* A special instance of sp 2051 applied to an equality with a dv condition. Unlike the more general sp 2051, we can prove this without ax-12 2045. Instance of aeveq 1980.

The antecedent 𝑥𝑥 = 𝑦 with distinct 𝑥 and 𝑦 is a characteristic of a degenerate universe, in which just one object exists. Actually more than one object may still exist, but if so, we give up on equality as a discriminating term.

Separating this degenerate case from a richer universe, where inequality is possible, is a common proof idea. The name of this theorem follows a convention, where the condition 𝑥𝑥 = 𝑦 is denoted by 'aev', a shorthand for 'all equal, with a distinct variable condition'. (Contributed by Wolf Lammen, 14-Mar-2021.)

(∀𝑥 𝑥 = 𝑦𝑥 = 𝑦)

Theoremcbvaev 1977* Change bound variable in an equality with a dv condition. Instance of aev 1981. (Contributed by NM, 22-Jul-2015.) (Revised by BJ, 18-Jun-2019.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑦)

Theoremaevlem0 1978* Lemma for aevlem 1979. Instance of aev 1981. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-12 2045. (Revised by Wolf Lammen, 14-Mar-2021.) (Revised by BJ, 29-Mar-2021.) (Proof shortened by Wolf Lammen, 30-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑥)

Theoremaevlem 1979* Lemma for aev 1981 and axc16g 2132. Change free and bound variables. Instance of aev 1981. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2244, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Revised by BJ, 29-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑧 = 𝑡)

Theoremaeveq 1980* The antecedent 𝑥𝑥 = 𝑦 with a dv condition (typical of a one-object universe) forces equality of everything. (Contributed by Wolf Lammen, 19-Mar-2021.)
(∀𝑥 𝑥 = 𝑦𝑧 = 𝑡)

Theoremaev 1981* A "distinctor elimination" lemma with no restrictions on variables in the consequent. (Contributed by NM, 8-Nov-2006.) Remove dependency on ax-11 2032. (Revised by Wolf Lammen, 7-Sep-2018.) Remove dependency on ax-13 2244, inspired by an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2045. (Revised by Wolf Lammen, 19-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧 𝑡 = 𝑢)

Theoremhbaevg 1982* Generalization of hbaev 1983, proved at no extra cost. Instance of aev2 1984. (Contributed by Wolf Lammen, 22-Mar-2021.) (Revised by BJ, 29-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑡 = 𝑢)

Theoremhbaev 1983* Version of hbae 2313 with a DV condition, requiring fewer axioms. Instance of hbaevg 1982 and aev2 1984. (Contributed by Wolf Lammen, 22-Mar-2021.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑥 𝑥 = 𝑦)

Theoremaev2 1984* A version of aev 1981 with two universal quantifiers in the consequent, and a generalization of hbaevg 1982. One can prove similar statements with arbitrary numbers of universal quantifiers in the consequent (the series begins with aeveq 1980, aev 1981, aev2 1984).

Using aev 1981 and alrimiv 1853 (as in aev2ALT 1985), one can actually prove (with no more axioms) any scheme of the form (∀𝑥𝑥 = 𝑦 PHI) , DV (𝑥, 𝑦) where PHI involves only setvar variables and the connectors , , , , , =, , , ∃*, ∃!, . An example is given by aevdemo 27287. This list cannot be extended to ¬ or since the scheme 𝑥𝑥 = 𝑦 is consistent with ax-mp 5, ax-gen 1720, ax-1 6-- ax-13 2244 (as the one-element universe shows).

(Contributed by BJ, 29-Mar-2021.)

(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)

Theoremaev2ALT 1985* Alternate proof of aev2 1984, bypassing hbaevg 1982. (Contributed by BJ, 23-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑥 𝑥 = 𝑦 → ∀𝑧𝑡 𝑢 = 𝑣)

Theoremaxc11nlemOLD2 1986* Lemma for axc11n 2305. Change bound variable in an equality. Obsolete as of 29-Mar-2021. Use aev 1981 instead. (Contributed by NM, 8-Jul-2016.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Restructure to ease either bundling, or reducing dependencies on axioms. (Revised by Wolf Lammen, 30-Nov-2019.) Remove dependency on ax-12 2045. (Revised by Wolf Lammen, 14-Mar-2021.) (Proof modification is discouraged.) (New usage is discouraged.)
(¬ ∀𝑦 𝑦 = 𝑥 → (𝑥 = 𝑧 → ∀𝑦 𝑥 = 𝑧))       (∀𝑥 𝑥 = 𝑧 → ∀𝑦 𝑦 = 𝑥)

TheoremaevlemOLD 1987* Old proof of aevlem 1979. Obsolete as of 29-Mar-2021. (Contributed by NM, 22-Jul-2015.) (Proof shortened by Wolf Lammen, 17-Feb-2018.) Remove dependency on ax-13 2244, along an idea of BJ. (Revised by Wolf Lammen, 30-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)
(∀𝑧 𝑧 = 𝑤 → ∀𝑦 𝑦 = 𝑥)

1.4.9  Membership predicate

Syntaxwcel 1988 Extend wff definition to include the membership connective between classes.

For a general discussion of the theory of classes, see mmset.html#class.

(The purpose of introducing wff 𝐴𝐵 here is to allow us to express i.e. "prove" the wel 1989 of predicate calculus in terms of the wcel 1988 of set theory, so that we don't "overload" the connective with two syntax definitions. This is done to prevent ambiguity that would complicate some Metamath parsers. The class variables 𝐴 and 𝐵 are introduced temporarily for the purpose of this definition but otherwise not used in predicate calculus. See df-clab 2607 for more information on the set theory usage of wcel 1988.)

wff 𝐴𝐵

Theoremwel 1989 Extend wff definition to include atomic formulas with the epsilon (membership) predicate. This is read "𝑥 is an element of 𝑦," "𝑥 is a member of 𝑦," "𝑥 belongs to 𝑦," or "𝑦 contains 𝑥." Note: The phrase "𝑦 includes 𝑥 " means "𝑥 is a subset of 𝑦;" to use it also for 𝑥𝑦, as some authors occasionally do, is poor form and causes confusion, according to George Boolos (1992 lecture at MIT).

This syntactic construction introduces a binary non-logical predicate symbol (epsilon) into our predicate calculus. We will eventually use it for the membership predicate of set theory, but that is irrelevant at this point: the predicate calculus axioms for apply to any arbitrary binary predicate symbol. "Non-logical" means that the predicate is presumed to have additional properties beyond the realm of predicate calculus, although these additional properties are not specified by predicate calculus itself but rather by the axioms of a theory (in our case set theory) added to predicate calculus. "Binary" means that the predicate has two arguments.

(Instead of introducing wel 1989 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 wcel 1988. This lets us avoid overloading the connective, thus preventing ambiguity that would complicate certain Metamath parsers. However, logically wel 1989 is considered to be a primitive syntax, even though here it is artificially "derived" from wcel 1988. Note: To see the proof steps of this syntax proof, type "show proof wel /all" in the Metamath program.) (Contributed by NM, 24-Jan-2006.)

wff 𝑥𝑦

1.4.10  Axiom scheme ax-8 (Left Equality for Binary Predicate)

Axiomax-8 1990 Axiom of Left Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the left-hand side of an arbitrary binary predicate , which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C12' in [Megill] p. 448 (p. 16 of the preprint). "Non-logical" means that the predicate is not a primitive of predicate calculus proper but instead is an extension to it. "Binary" means that the predicate has two arguments. In a system of predicate calculus with equality, like ours, equality is not usually considered to be a non-logical predicate. In systems of predicate calculus without equality, it typically would be.

We prove in ax8 1994 that this axiom can be recovered from its weakened version ax8v 1991 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-8 1990 should be ax8v 1991. See the comment of ax8v 1991 for more details on these matters. (Contributed by NM, 30-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax8 1994 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremax8v 1991* Weakened version of ax-8 1990, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-8 1990, and it should be referenced only by its two weakened versions ax8v1 1992 and ax8v2 1993, from which ax-8 1990 is then rederived as ax8 1994, which shows that either ax8v 1991 or the conjunction of ax8v1 1992 and ax8v2 1993 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax8 1994 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremax8v1 1992* First of two weakened versions of ax8v 1991, with an extra dv condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremax8v2 1993* Second of two weakened versions of ax8v 1991, with an extra dv condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremax8 1994 Proof of ax-8 1990 from ax8v1 1992 and ax8v2 1993, proving sufficiency of the conjunction of the latter two weakened versions of ax8v 1991, which is itself a weakened version of ax-8 1990. (Contributed by BJ, 7-Dec-2020.) (Proof shortened by Wolf Lammen, 11-Apr-2021.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremelequ1 1995 An identity law for the non-logical predicate. (Contributed by NM, 30-Jun-1993.)
(𝑥 = 𝑦 → (𝑥𝑧𝑦𝑧))

Theoremcleljust 1996* When the class variables in definition df-clel 2616 are replaced with setvar variables, this theorem of predicate calculus is the result. This theorem provides part of the justification for the consistency of that definition, which "overloads" the setvar variables in wel 1989 with the class variables in wcel 1988. (Contributed by NM, 28-Jan-2004.) Revised to use equsexvw 1930 in order to remove dependencies on ax-10 2017, ax-12 2045, ax-13 2244. Note that there is no DV condition on 𝑥, 𝑦, that is, on the variables of the left-hand side. (Revised by BJ, 29-Dec-2020.)
(𝑥𝑦 ↔ ∃𝑧(𝑧 = 𝑥𝑧𝑦))

1.4.11  Axiom scheme ax-9 (Right Equality for Binary Predicate)

Axiomax-9 1997 Axiom of Right Equality for Binary Predicate. One of the equality and substitution axioms for a non-logical predicate in our predicate calculus with equality. It substitutes equal variables into the right-hand side of an arbitrary binary predicate , which we will use for the set membership relation when set theory is introduced. This axiom scheme is a sub-scheme of Axiom Scheme B8 of system S2 of [Tarski], p. 75, whose general form cannot be represented with our notation. Also appears as Axiom scheme C13' in [Megill] p. 448 (p. 16 of the preprint).

We prove in ax9 2001 that this axiom can be recovered from its weakened version ax9v 1998 where 𝑥 and 𝑦 are assumed to be disjoint variables. In particular, the only theorem referencing ax-9 1997 should be ax9v 1998. See the comment of ax9v 1998 for more details on these matters. (Contributed by NM, 21-Jun-1993.) (Revised by BJ, 7-Dec-2020.) Use ax9 2001 instead. (New usage is discouraged.)

(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremax9v 1998* Weakened version of ax-9 1997, with a dv condition on 𝑥, 𝑦. This should be the only proof referencing ax-9 1997, and it should be referenced only by its two weakened versions ax9v1 1999 and ax9v2 2000, from which ax-9 1997 is then rederived as ax9 2001, which shows that either ax9v 1998 or the conjunction of ax9v1 1999 and ax9v2 2000 is sufficient. (Contributed by BJ, 7-Dec-2020.) Use ax9 2001 instead. (New usage is discouraged.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremax9v1 1999* First of two weakened versions of ax9v 1998, with an extra dv condition on 𝑥, 𝑧, see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

Theoremax9v2 2000* Second of two weakened versions of ax9v 1998, with an extra dv condition on 𝑦, 𝑧 see comments there. (Contributed by BJ, 7-Dec-2020.)
(𝑥 = 𝑦 → (𝑧𝑥𝑧𝑦))

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392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42316
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