P. 439 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 43801-43900   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	frege20 43801	A closed form of syl8 76. Proposition 20 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜃 → 𝜏) → (𝜑 → (𝜓 → (𝜒 → 𝜏)))))
21.36.6.4  _Begriffsschrift_ Chapter II Implication and Negation
Theorem	axfrege28 43802	Contraposition. Identical to con3 153. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
Axiom	ax-frege28 43803	Contraposition. Identical to con3 153. Theorem *2.16 of [WhiteheadRussell] p. 103. Axiom 28 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
Theorem	frege29 43804	Closed form of con3d 152. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
Theorem	frege30 43805	Commuted, closed form of con3d 152. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))
Theorem	axfrege31 43806	Identical to notnotr 130. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
⊢ (¬ ¬ 𝜑 → 𝜑)
Axiom	ax-frege31 43807	𝜑 cannot be denied and (at the same time ) ¬ ¬ 𝜑 affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnotr 130. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ (¬ ¬ 𝜑 → 𝜑)
Theorem	frege32 43808	Deduce con1 146 from con3 153. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)))
Theorem	frege33 43809	If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 146. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑))
Theorem	frege34 43810	If as a consequence of the occurrence of the circumstance 𝜑, when the obstacle 𝜓 is removed, 𝜒 takes place, then from the circumstance that 𝜒 does not take place while 𝜑 occurs the occurrence of the obstacle 𝜓 can be inferred. Closed form of con1d 145. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → 𝜓)))
Theorem	frege35 43811	Commuted, closed form of con1d 145. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → (¬ 𝜒 → (𝜑 → 𝜓)))
Theorem	frege36 43812	The case in which 𝜓 is denied, ¬ 𝜑 is affirmed, and 𝜑 is affirmed does not occur. If 𝜑 occurs, then (at least) one of the two, 𝜑 or 𝜓, takes place (no matter what 𝜓 might be). Identical to pm2.24 124. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → (¬ 𝜑 → 𝜓))
Theorem	frege37 43813	If 𝜒 is a necessary consequence of the occurrence of 𝜓 or 𝜑, then 𝜒 is a necessary consequence of 𝜑 alone. Similar to a closed form of orcs 875. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((¬ 𝜑 → 𝜓) → 𝜒) → (𝜑 → 𝜒))
Theorem	frege38 43814	Identical to pm2.21 123. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (¬ 𝜑 → (𝜑 → 𝜓))
Theorem	frege39 43815	Syllogism between pm2.18 128 and pm2.24 124. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → (¬ 𝜑 → 𝜓))
Theorem	frege40 43816	Anything implies pm2.18 128. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (¬ 𝜑 → ((¬ 𝜓 → 𝜓) → 𝜓))
Theorem	axfrege41 43817	Identical to notnot 142. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ¬ ¬ 𝜑)
Axiom	ax-frege41 43818	The affirmation of 𝜑 denies the denial of 𝜑. Identical to notnot 142. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ (𝜑 → ¬ ¬ 𝜑)
Theorem	frege42 43819	Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ¬ ¬ (𝜑 → 𝜑)
Theorem	frege43 43820	If there is a choice only between 𝜑 and 𝜑, then 𝜑 takes place. Identical to pm2.18 128. Proposition 43 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜑) → 𝜑)
Theorem	frege44 43821	Similar to a commuted pm2.62 899. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜑))
Theorem	frege45 43822	Deduce pm2.6 191 from con1 146. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((¬ 𝜑 → 𝜓) → (¬ 𝜓 → 𝜑)) → ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)))
Theorem	frege46 43823	If 𝜓 holds when 𝜑 occurs as well as when 𝜑 does not occur, then 𝜓 holds. If 𝜓 or 𝜑 occurs and if the occurrences of 𝜑 has 𝜓 as a necessary consequence, then 𝜓 takes place. Identical to pm2.6 191. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓))
Theorem	frege47 43824	Deduce consequence follows from either path implied by a disjunction. If 𝜑, as well as 𝜓 is sufficient condition for 𝜒 and 𝜓 or 𝜑 takes place, then the proposition 𝜒 holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜒) → ((𝜑 → 𝜒) → 𝜒)))
Theorem	frege48 43825	Closed form of syllogism with internal disjunction. If 𝜑 is a sufficient condition for the occurrence of 𝜒 or 𝜓 and if 𝜒, as well as 𝜓, is a sufficient condition for 𝜃, then 𝜑 is a sufficient condition for 𝜃. See application in frege101 43937. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (¬ 𝜓 → 𝜒)) → ((𝜒 → 𝜃) → ((𝜓 → 𝜃) → (𝜑 → 𝜃))))
Theorem	frege49 43826	Closed form of deduction with disjunction. Proposition 49 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜒) → ((𝜓 → 𝜒) → 𝜒)))
Theorem	frege50 43827	Closed form of jaoi 857. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜓) → ((¬ 𝜑 → 𝜒) → 𝜓)))
Theorem	frege51 43828	Compare with jaod 859. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜒) → (𝜑 → ((¬ 𝜓 → 𝜃) → 𝜒))))
21.36.6.5  _Begriffsschrift_ Chapter II with logical equivalence Here we leverage df-ifp 1063 to partition a wff into two that are disjoint with the selector wff. Thus if we are given ⊢ (𝜑 ↔ if-(𝜓, 𝜒, 𝜃)) then we replace the concept (illegal in our notation ) (𝜑‘𝜓) with if-(𝜓, 𝜒, 𝜃) to reason about the values of the "function." Likewise, we replace the similarly illegal concept ∀𝜓𝜑 with (𝜒 ∧ 𝜃).
Theorem	axfrege52a 43829	Justification for ax-frege52a 43830. (Contributed by RP, 17-Apr-2020.)
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
Axiom	ax-frege52a 43830	The case when the content of 𝜑 is identical with the content of 𝜓 and in which a proposition controlled by an element for which we substitute the content of 𝜑 is affirmed (in this specific case the identity logical function) and the same proposition, this time where we substituted the content of 𝜓, is denied does not take place. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
Theorem	frege52aid 43831	The case when the content of 𝜑 is identical with the content of 𝜓 and in which 𝜑 is affirmed and 𝜓 is denied does not take place. Identical to biimp 215. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓))
Theorem	frege53aid 43832	Specialization of frege53a 43833. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ((𝜑 ↔ 𝜓) → 𝜓))
Theorem	frege53a 43833	Lemma for frege55a 43841. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (if-(𝜑, 𝜃, 𝜒) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜃, 𝜒)))
Theorem	axfrege54a 43834	Justification for ax-frege54a 43835. Identical to biid 261. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 ↔ 𝜑)
Axiom	ax-frege54a 43835	Reflexive equality of wffs. The content of 𝜑 is identical with the content of 𝜑. Part of Axiom 54 of [Frege1879] p. 50. Identical to biid 261. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ (𝜑 ↔ 𝜑)
Theorem	frege54cor0a 43836	Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
Theorem	frege54cor1a 43837	Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ if-(𝜑, 𝜑, ¬ 𝜑)
Theorem	frege55aid 43838	Lemma for frege57aid 43845. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
Theorem	frege55lem1a 43839	Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜏 → if-(𝜓, 𝜑, ¬ 𝜑)) → (𝜏 → (𝜓 ↔ 𝜑)))
Theorem	frege55lem2a 43840	Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))
Theorem	frege55a 43841	Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))
Theorem	frege55cor1a 43842	Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
Theorem	frege56aid 43843	Lemma for frege57aid 43845. 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 43844	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 43845	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 43846	Analogue of frege57aid 43845. 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 43847	Identical to anifp 1071. Justification for ax-frege58a 43848. (Contributed by RP, 28-Mar-2020.)
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
Axiom	ax-frege58a 43848	If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2069. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒))
Theorem	frege58acor 43849	Lemma for frege59a 43850. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
⊢ (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
Theorem	frege59a 43850	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 43786 incorrectly referenced where frege30 43805 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 43851	Swap antecedents of ax-frege58a 43848. 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 43852	Lemma for frege65a 43856. 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 43853	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2656 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 43854	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 43855	Lemma for frege65a 43856. 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 43856	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2656 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 43857	Swap antecedents of frege65a 43856. 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 43858	Lemma for frege68a 43859. 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 43859	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 43860	Justification for ax-frege52c 43861. (Contributed by RP, 24-Dec-2019.)
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑))
Axiom	ax-frege52c 43861	One side of dfsbcq 3746. 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 43862	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 43863	Lemma for frege102 (via frege92 43928). 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 43864	Reflexive equality of classes. Identical to eqid 2729. Justification for ax-frege54c 43865. (Contributed by RP, 24-Dec-2019.)
⊢ 𝐴 = 𝐴
Axiom	ax-frege54c 43865	Reflexive equality of sets (as classes). Part of Axiom 54 of [Frege1879] p. 50. Identical to eqid 2729. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ 𝐴 = 𝐴
Theorem	frege54b 43866	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 2729. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝑥 = 𝑥
Theorem	frege54cor1b 43867	Reflexive equality. (Contributed by RP, 24-Dec-2019.)
⊢ [𝑥 / 𝑦]𝑦 = 𝑥
Theorem	frege55lem1b 43868*	Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑 → 𝑥 = 𝑧))
Theorem	frege55lem2b 43869	Lemma for frege55b 43870. 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 43870	Lemma for frege57b 43872. Proposition 55 of [Frege1879] p. 50. Note that eqtr2 2750 incorporates eqcom 2736 which is stronger than this proposition which is identical to equcomi 2017. 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 43871	Lemma for frege57b 43872. 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 43872	Analogue of frege57aid 43845. 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 43873	If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2069. Justification for ax-frege58b 43874. (Contributed by RP, 28-Mar-2020.)
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
Axiom	ax-frege58b 43874	If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2069. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑)
Theorem	frege58bid 43875	If ∀𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2184. See ax-frege58b 43874 and frege58c 43894 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 43876	Lemma for frege59b 43877. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓))
Theorem	frege59b 43877	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 43786 incorrectly referenced where frege30 43805 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 43878	Swap antecedents of ax-frege58b 43874. 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 43879	Lemma for frege65b 43883. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (([𝑥 / 𝑦]𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓))
Theorem	frege62b 43880	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2656 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 43881	Lemma for frege91 43927. 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 43882	Lemma for frege65b 43883. 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 43883	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2656 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 43884	Swap antecedents of frege65b 43883. 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 43885	Lemma for frege68b 43886. 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 43886	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 43887	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 43888*	Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)
⊢ 𝐴 ∈ 𝐶    ⇒   ⊢ [𝐴 / 𝑥]𝑥 = 𝐴
Theorem	frege55lem1c 43889*	Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 𝐵))
Theorem	frege55lem2c 43890*	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 43891	Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥)
Theorem	frege56c 43892*	Lemma for frege57c 43893. 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 43893*	Swap order of implication in ax-frege52c 43861. 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 43894	Principle related to sp 2184. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ 𝐴 ∈ 𝐵    ⇒   ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑)
Theorem	frege59c 43895	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 43786 incorrectly referenced where frege30 43805 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 43896	Swap antecedents of frege58c 43894. 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 43897	Lemma for frege65c 43901. 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 43898	A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2656 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 43899	Analogue of frege63b 43881. 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 43900	Lemma for frege65c 43901. 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)))