P. 440 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43901-44000   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	frege55cor1a 43901	Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
Theorem	frege56aid 43902	Lemma for frege57aid 43904. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) → ((𝜓 ↔ 𝜑) → (𝜑 → 𝜓)))
Theorem	frege56a 43903	Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓 ↔ 𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))))
Theorem	frege57aid 43904	This is the all important formula which allows to apply Frege-style definitions and explore their consequences. A closed form of biimpri 228. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))
Theorem	frege57a 43905	Analogue of frege57aid 43904. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))
Theorem	axfrege58a 43906	Identical to anifp 1071. Justification for ax-frege58a 43907. (Contributed by RP, 28-Mar-2020.)
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
Axiom	ax-frege58a 43907	If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2071. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
Theorem	frege58acor 43908	Lemma for frege59a 43909. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
Theorem	frege59a 43909	A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of [Frege1879] p. 51. Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 43845 incorrectly referenced where frege30 43864 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ (if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓 → 𝜒) ∧ (𝜃 → 𝜏))))
Theorem	frege60a 43910	Swap antecedents of ax-frege58a 43907. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
Theorem	frege61a 43911	Lemma for frege65a 43915. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ ((if-(𝜑, 𝜓, 𝜒) → 𝜃) → ((𝜓 ∧ 𝜒) → 𝜃))
Theorem	frege62a 43912	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2658 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ (if-(𝜑, 𝜓, 𝜃) → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏)))
Theorem	frege63a 43913	Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ (if-(𝜑, 𝜓, 𝜃) → (𝜂 → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏))))
Theorem	frege64a 43914	Lemma for frege65a 43915. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁))))
Theorem	frege65a 43915	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2658 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
Theorem	frege66a 43916	Swap antecedents of frege65a 43915. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
Theorem	frege67a 43917	Lemma for frege68a 43918. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ ((((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → (𝜓 ∧ 𝜒))) → (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))))
Theorem	frege68a 43918	Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ (((𝜓 ∧ 𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))
21.36.6.6  _Begriffsschrift_ Chapter II with equivalence of sets
Theorem	axfrege52c 43919	Justification for ax-frege52c 43920. (Contributed by RP, 24-Dec-2019.)
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))
Axiom	ax-frege52c 43920	One side of dfsbcq 3743. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))
Theorem	frege52b 43921	The case when the content of 𝑥 is identical with the content of 𝑦 and in which a proposition controlled by an element for which we substitute the content of 𝑥 is affirmed and the same proposition, this time where we substitute the content of 𝑦, is denied does not take place. In [𝑥 / 𝑧]𝜑, 𝑥 can also occur in other than the argument (𝑧) places. Hence 𝑥 may still be contained in [𝑦 / 𝑧]𝜑. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑))
Theorem	frege53b 43922	Lemma for frege102 (via frege92 43987). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → (𝑦 = 𝑧 → [𝑧 / 𝑥]𝜑))
Theorem	axfrege54c 43923	Reflexive equality of classes. Identical to eqid 2731. Justification for ax-frege54c 43924. (Contributed by RP, 24-Dec-2019.)
⊢ 𝐴 = 𝐴
Axiom	ax-frege54c 43924	Reflexive equality of sets (as classes). Part of Axiom 54 of [Frege1879] p. 50. Identical to eqid 2731. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ 𝐴 = 𝐴
Theorem	frege54b 43925	Reflexive equality of sets. The content of 𝑥 is identical with the content of 𝑥. Part of Axiom 54 of [Frege1879] p. 50. Slightly specialized version of eqid 2731. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝑥 = 𝑥
Theorem	frege54cor1b 43926	Reflexive equality. (Contributed by RP, 24-Dec-2019.)
⊢ [𝑥 / 𝑦]𝑦 = 𝑥
Theorem	frege55lem1b 43927*	Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑 → 𝑥 = 𝑧))
Theorem	frege55lem2b 43928	Lemma for frege55b 43929. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → [𝑦 / 𝑧]𝑧 = 𝑥)
Theorem	frege55b 43929	Lemma for frege57b 43931. Proposition 55 of [Frege1879] p. 50. Note that eqtr2 2752 incorporates eqcom 2738 which is stronger than this proposition which is identical to equcomi 2018. Is it possible that Frege tricked himself into assuming what he was out to prove? (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → 𝑦 = 𝑥)
Theorem	frege56b 43930	Lemma for frege57b 43931. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝑥 = 𝑦 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)) → (𝑦 = 𝑥 → ([𝑥 / 𝑧]𝜑 → [𝑦 / 𝑧]𝜑)))
Theorem	frege57b 43931	Analogue of frege57aid 43904. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝑦 → ([𝑦 / 𝑧]𝜑 → [𝑥 / 𝑧]𝜑))
Theorem	axfrege58b 43932	If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2071. Justification for ax-frege58b 43933. (Contributed by RP, 28-Mar-2020.)
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
Axiom	ax-frege58b 43933	If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2071. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
Theorem	frege58bid 43934	If ∀𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2186. See ax-frege58b 43933 and frege58c 43953 for versions which more closely track the original. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥𝜑 → 𝜑)
Theorem	frege58bcor 43935	Lemma for frege59b 43936. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	frege59b 43936	A kind of Aristotelian inference. Namely Felapton or Fesapo. Proposition 59 of [Frege1879] p. 51. Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 43845 incorrectly referenced where frege30 43864 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → (¬ [𝑦 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓)))
Theorem	frege60b 43937	Swap antecedents of ax-frege58b 43933. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝑦 / 𝑥]𝜓 → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
Theorem	frege61b 43938	Lemma for frege65b 43942. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (([𝑥 / 𝑦]𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓))
Theorem	frege62b 43939	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2658 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝑦 / 𝑥]𝜓))
Theorem	frege63b 43940	Lemma for frege91 43986. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝑦 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑 → 𝜒) → [𝑦 / 𝑥]𝜒)))
Theorem	frege64b 43941	Lemma for frege65b 43942. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜓) → (∀𝑦(𝜓 → 𝜒) → ([𝑥 / 𝑦]𝜑 → [𝑧 / 𝑦]𝜒)))
Theorem	frege65b 43942	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2658 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. In Frege care is taken to point out that the variables in the first clauses are independent of each other and of the final term so another valid translation could be : ⊢ (∀𝑥([𝑥 / 𝑎]𝜑 → [𝑥 / 𝑏]𝜓) → (∀𝑦([𝑦 / 𝑏]𝜓 → [𝑦 / 𝑐]𝜒) → ([𝑧 / 𝑎]𝜑 → [𝑧 / 𝑐]𝜒))). But that is perhaps too pedantic a translation for this exploration. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜒)))
Theorem	frege66b 43943	Swap antecedents of frege65b 43942. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝑦 / 𝑥]𝜒 → [𝑦 / 𝑥]𝜓)))
Theorem	frege67b 43944	Lemma for frege68b 43945. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑)))
Theorem	frege68b 43945	Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝑦 / 𝑥]𝜑))
21.36.6.7  _Begriffsschrift_ Chapter II with equivalence of classes Begriffsschrift Chapter II with equivalence of classes (where they are sets).
Theorem	frege53c 43946	Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ([𝐴 / 𝑥]𝜑 → (𝐴 = 𝐵 → [𝐵 / 𝑥]𝜑))
Theorem	frege54cor1c 43947*	Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)
⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ [𝐴 / 𝑥]𝑥 = 𝐴
Theorem	frege55lem1c 43948*	Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 𝐵))
Theorem	frege55lem2c 43949*	Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝐴 → [𝐴 / 𝑧]𝑧 = 𝑥)
Theorem	frege55c 43950	Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥)
Theorem	frege56c 43951*	Lemma for frege57c 43952. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐵 ∈ 𝐶    ⇒   ⊢ ((𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) → (𝐵 = 𝐴 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)))
Theorem	frege57c 43952*	Swap order of implication in ax-frege52c 43920. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ (𝐴 = 𝐵 → ([𝐵 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜑))
Theorem	frege58c 43953	Principle related to sp 2186. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)
Theorem	frege59c 43954	A kind of Aristotelian inference. Proposition 59 of [Frege1879] p. 51. Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 43845 incorrectly referenced where frege30 43864 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 → (¬ [𝐴 / 𝑥]𝜓 → ¬ ∀𝑥(𝜑 → 𝜓)))
Theorem	frege60c 43955	Swap antecedents of frege58c 43953. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (∀𝑥(𝜑 → (𝜓 → 𝜒)) → ([𝐴 / 𝑥]𝜓 → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))
Theorem	frege61c 43956	Lemma for frege65c 43960. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (([𝐴 / 𝑥]𝜑 → 𝜓) → (∀𝑥𝜑 → 𝜓))
Theorem	frege62c 43957	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2658 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 → (∀𝑥(𝜑 → 𝜓) → [𝐴 / 𝑥]𝜓))
Theorem	frege63c 43958	Analogue of frege63b 43940. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ([𝐴 / 𝑥]𝜑 → (𝜓 → (∀𝑥(𝜑 → 𝜒) → [𝐴 / 𝑥]𝜒)))
Theorem	frege64c 43959	Lemma for frege65c 43960. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐶 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))
Theorem	frege65c 43960	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2658 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜓 → 𝜒) → ([𝐴 / 𝑥]𝜑 → [𝐴 / 𝑥]𝜒)))
Theorem	frege66c 43961	Swap antecedents of frege65c 43960. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (∀𝑥(𝜑 → 𝜓) → (∀𝑥(𝜒 → 𝜑) → ([𝐴 / 𝑥]𝜒 → [𝐴 / 𝑥]𝜓)))
Theorem	frege67c 43962	Lemma for frege68c 43963. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → ∀𝑥𝜑)) → ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑)))
Theorem	frege68c 43963	Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ((∀𝑥𝜑 ↔ 𝜓) → (𝜓 → [𝐴 / 𝑥]𝜑))
21.36.6.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence (𝑅 “ 𝐴) ⊆ 𝐴 means membership in 𝐴 is hereditary in the sequence dictated by relation 𝑅. This differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on. While the above notation is modern, it is cumbersome in the case when 𝐴 is complex and to more closely follow Frege, we abbreviate it with new notation 𝑅 hereditary 𝐴. This greatly shortens the statements for frege97 43992 and frege109 44004. dffrege69 43964 through frege75 43970 develop this, but translation to Metamath is pending some decisions. While Frege does not limit discussion to sets, we may have to depart from Frege by limiting 𝑅 or 𝐴 to sets when we quantify over all hereditary relations or all classes where membership is hereditary in a sequence dictated by 𝑅.
Theorem	dffrege69 43964*	If from the proposition that 𝑥 has property 𝐴 it can be inferred generally, whatever 𝑥 may be, that every result of an application of the procedure 𝑅 to 𝑥 has property 𝐴, then we say " Property 𝐴 is hereditary in the 𝑅-sequence. Definition 69 of [Frege1879] p. 55. (Contributed by RP, 28-Mar-2020.)
⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) ↔ 𝑅 hereditary 𝐴)
Theorem	frege70 43965*	Lemma for frege72 43967. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑉    ⇒   ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → ∀𝑦(𝑋𝑅𝑦 → 𝑦 ∈ 𝐴)))
Theorem	frege71 43966*	Lemma for frege72 43967. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑉    ⇒   ⊢ ((∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴))))
Theorem	frege72 43967	If property 𝐴 is hereditary in the 𝑅-sequence, if 𝑥 has property 𝐴, and if 𝑦 is a result of an application of the procedure 𝑅 to 𝑥, then 𝑦 has property 𝐴. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege73 43968	Lemma for frege87 43982. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ ((𝑅 hereditary 𝐴 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege74 43969	If 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then every result of a application of the procedure 𝑅 to 𝑋 has the property 𝐴. Proposition 74 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    ⇒   ⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋𝑅𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege75 43970*	If from the proposition that 𝑥 has property 𝐴, whatever 𝑥 may be, it can be inferred that every result of an application of the procedure 𝑅 to 𝑥 has property 𝐴, then property 𝐴 is hereditary in the 𝑅-sequence. Proposition 75 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)) → 𝑅 hereditary 𝐴)
21.36.6.9  _Begriffsschrift_ Chapter III Following in a sequence 𝑝(t+‘𝑅)𝑐 means 𝑐 follows 𝑝 in the 𝑅-sequence. dffrege76 43971 through frege98 43993 develop this. This will be shown to be the transitive closure of the relation 𝑅. But more work needs to be done on transitive closure of relations before this is ready for Metamath.
Theorem	dffrege76 43971*	If from the two propositions that every result of an application of the procedure 𝑅 to 𝐵 has property 𝑓 and that property 𝑓 is hereditary in the 𝑅-sequence, it can be inferred, whatever 𝑓 may be, that 𝐸 has property 𝑓, then we say 𝐸 follows 𝐵 in the 𝑅-sequence. Definition 76 of [Frege1879] p. 60. Each of 𝐵, 𝐸 and 𝑅 must be sets. (Contributed by RP, 2-Jul-2020.)
⊢ 𝐵 ∈ 𝑈    &   ⊢ 𝐸 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑎(𝐵𝑅𝑎 → 𝑎 ∈ 𝑓) → 𝐸 ∈ 𝑓)) ↔ 𝐵(t+‘𝑅)𝐸)
Theorem	frege77 43972*	If 𝑌 follows 𝑋 in the 𝑅-sequence, if property 𝐴 is hereditary in the 𝑅-sequence, and if every result of an application of the procedure 𝑅 to 𝑋 has the property 𝐴, then 𝑌 has property 𝐴. Proposition 77 of [Frege1879] p. 62. (Contributed by RP, 29-Jun-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → 𝑌 ∈ 𝐴)))
Theorem	frege78 43973*	Commuted form of frege77 43972. Proposition 78 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑅 hereditary 𝐴 → (∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴) → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege79 43974*	Distributed form of frege78 43973. Proposition 79 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ((𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴)) → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege80 43975*	Add additional condition to both clauses of frege79 43974. Proposition 80 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ((𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → ∀𝑎(𝑋𝑅𝑎 → 𝑎 ∈ 𝐴))) → (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴))))
Theorem	frege81 43976	If 𝑋 has a property 𝐴 that is hereditary in the 𝑅 -sequence, and if 𝑌 follows 𝑋 in the 𝑅-sequence, then 𝑌 has property 𝐴. This is a form of induction attributed to Jakob Bernoulli. Proposition 81 of [Frege1879] p. 63. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋 ∈ 𝐴 → (𝑅 hereditary 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege82 43977	Closed-form deduction based on frege81 43976. Proposition 82 of [Frege1879] p. 64. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ ((𝜑 → 𝑋 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝜑 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴))))
Theorem	frege83 43978	Apply commuted form of frege81 43976 when the property 𝑅 is hereditary in a disjunction of two properties, only one of which is known to be held by 𝑋. Proposition 83 of [Frege1879] p. 65. Here we introduce the union of classes where Frege has a disjunction of properties which are represented by membership in either of the classes. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑆    &   ⊢ 𝑌 ∈ 𝑇    &   ⊢ 𝑅 ∈ 𝑈    &   ⊢ 𝐵 ∈ 𝑉    &   ⊢ 𝐶 ∈ 𝑊    ⇒   ⊢ (𝑅 hereditary (𝐵 ∪ 𝐶) → (𝑋 ∈ 𝐵 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ (𝐵 ∪ 𝐶))))
Theorem	frege84 43979	Commuted form of frege81 43976. Proposition 84 of [Frege1879] p. 65. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑅 hereditary 𝐴 → (𝑋 ∈ 𝐴 → (𝑋(t+‘𝑅)𝑌 → 𝑌 ∈ 𝐴)))
Theorem	frege85 43980*	Commuted form of frege77 43972. Proposition 85 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝐴) → (𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴)))
Theorem	frege86 43981*	Conclusion about element one past 𝑌 in the 𝑅-sequence. Proposition 86 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (((𝑅 hereditary 𝐴 → 𝑌 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴))) → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴)))))
Theorem	frege87 43982*	If 𝑍 is a result of an application of the procedure 𝑅 to an object 𝑌 that follows 𝑋 in the 𝑅-sequence and if every result of an application of the procedure 𝑅 to 𝑋 has a property 𝐴 that is hereditary in the 𝑅-sequence, then 𝑍 has property 𝐴. Proposition 87 of [Frege1879] p. 66. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → (𝑌𝑅𝑍 → 𝑍 ∈ 𝐴))))
Theorem	frege88 43983*	Commuted form of frege87 43982. Proposition 88 of [Frege1879] p. 67. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝑆    &   ⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝐴) → (𝑅 hereditary 𝐴 → 𝑍 ∈ 𝐴))))
Theorem	frege89 43984*	One direction of dffrege76 43971. Proposition 89 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → 𝑌 ∈ 𝑓)) → 𝑋(t+‘𝑅)𝑌)
Theorem	frege90 43985*	Add antecedent to frege89 43984. Proposition 90 of [Frege1879] p. 68. (Contributed by RP, 1-Jul-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ ((𝜑 → ∀𝑓(𝑅 hereditary 𝑓 → (∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → 𝑌 ∈ 𝑓))) → (𝜑 → 𝑋(t+‘𝑅)𝑌))
Theorem	frege91 43986	Every result of an application of a procedure 𝑅 to an object 𝑋 follows that 𝑋 in the 𝑅-sequence. Proposition 91 of [Frege1879] p. 68. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (𝑋𝑅𝑌 → 𝑋(t+‘𝑅)𝑌)
Theorem	frege92 43987	Inference from frege91 43986. Proposition 92 of [Frege1879] p. 69. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (𝑋 = 𝑍 → (𝑋𝑅𝑌 → 𝑍(t+‘𝑅)𝑌))
Theorem	frege93 43988*	Necessary condition for two elements to be related by the transitive closure. Proposition 93 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ (∀𝑓(∀𝑧(𝑋𝑅𝑧 → 𝑧 ∈ 𝑓) → (𝑅 hereditary 𝑓 → 𝑌 ∈ 𝑓)) → 𝑋(t+‘𝑅)𝑌)
Theorem	frege94 43989*	Looking one past a pair related by transitive closure of a relation. Proposition 94 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑍 ∈ 𝑉    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ ((𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → ∀𝑓(∀𝑤(𝑋𝑅𝑤 → 𝑤 ∈ 𝑓) → (𝑅 hereditary 𝑓 → 𝑍 ∈ 𝑓)))) → (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → 𝑋(t+‘𝑅)𝑍)))
Theorem	frege95 43990	Looking one past a pair related by transitive closure of a relation. Proposition 95 of [Frege1879] p. 70. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝐴    ⇒   ⊢ (𝑌𝑅𝑍 → (𝑋(t+‘𝑅)𝑌 → 𝑋(t+‘𝑅)𝑍))
Theorem	frege96 43991	Every result of an application of the procedure 𝑅 to an object that follows 𝑋 in the 𝑅-sequence follows 𝑋 in the 𝑅 -sequence. Proposition 96 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑌 ∈ 𝑉    &   ⊢ 𝑍 ∈ 𝑊    &   ⊢ 𝑅 ∈ 𝐴    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌𝑅𝑍 → 𝑋(t+‘𝑅)𝑍))
Theorem	frege97 43992	The property of following 𝑋 in the 𝑅-sequence is hereditary in the 𝑅-sequence. Proposition 97 of [Frege1879] p. 71. Here we introduce the image of a singleton under a relation as class which stands for the property of following 𝑋 in the 𝑅 -sequence. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝑈    &   ⊢ 𝑅 ∈ 𝑊    ⇒   ⊢ 𝑅 hereditary ((t+‘𝑅) “ {𝑋})
Theorem	frege98 43993	If 𝑌 follows 𝑋 and 𝑍 follows 𝑌 in the 𝑅-sequence then 𝑍 follows 𝑋 in the 𝑅-sequence because the transitive closure of a relation has the transitive property. Proposition 98 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.) (Revised by RP, 6-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝐴    &   ⊢ 𝑌 ∈ 𝐵    &   ⊢ 𝑍 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑋(t+‘𝑅)𝑌 → (𝑌(t+‘𝑅)𝑍 → 𝑋(t+‘𝑅)𝑍))
21.36.6.10  _Begriffsschrift_ Chapter III Member of sequence 𝑝((t+‘𝑅) ∪ I )𝑐 means 𝑐 is a member of the 𝑅 -sequence beginning with 𝑝 and 𝑝 is a member of the 𝑅 -sequence ending with 𝑐. dffrege99 43994 through frege114 44009 develop this. This will be shown to be related to the transitive-reflexive closure of relation 𝑅. But more work needs to be done on transitive closure of relations before this is ready for Metamath.
Theorem	dffrege99 43994	If 𝑍 is identical with 𝑋 or follows 𝑋 in the 𝑅 -sequence, then we say : "𝑍 belongs to the 𝑅-sequence beginning with 𝑋 " or "𝑋 belongs to the 𝑅-sequence ending with 𝑍". Definition 99 of [Frege1879] p. 71. (Contributed by RP, 2-Jul-2020.)
⊢ 𝑍 ∈ 𝑈    ⇒   ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) ↔ 𝑋((t+‘𝑅) ∪ I )𝑍)
Theorem	frege100 43995	One direction of dffrege99 43994. Proposition 100 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑈    ⇒   ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋))
Theorem	frege101 43996	Lemma for frege102 43997. Proposition 101 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑈    ⇒   ⊢ ((𝑍 = 𝑋 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → ((𝑋(t+‘𝑅)𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉)) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉))))
Theorem	frege102 43997	If 𝑍 belongs to the 𝑅-sequence beginning with 𝑋, then every result of an application of the procedure 𝑅 to 𝑍 follows 𝑋 in the 𝑅-sequence. Proposition 102 of [Frege1879] p. 72. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑋 ∈ 𝐴    &   ⊢ 𝑍 ∈ 𝐵    &   ⊢ 𝑉 ∈ 𝐶    &   ⊢ 𝑅 ∈ 𝐷    ⇒   ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (𝑍𝑅𝑉 → 𝑋(t+‘𝑅)𝑉))
Theorem	frege103 43998	Proposition 103 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ ((𝑍 = 𝑋 → 𝑋 = 𝑍) → (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍)))
Theorem	frege104 43999	Proposition 104 of [Frege1879] p. 73. Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the minor clause and result swapped. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ (𝑋((t+‘𝑅) ∪ I )𝑍 → (¬ 𝑋(t+‘𝑅)𝑍 → 𝑋 = 𝑍))
Theorem	frege105 44000	Proposition 105 of [Frege1879] p. 73. (Contributed by RP, 7-Jul-2020.) (Proof modification is discouraged.)
⊢ 𝑍 ∈ 𝑉    ⇒   ⊢ ((¬ 𝑋(t+‘𝑅)𝑍 → 𝑍 = 𝑋) → 𝑋((t+‘𝑅) ∪ I )𝑍)