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Type | Label | Description |
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Statement | ||
Theorem | frege43 42901 | 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 42902 | Similar to a commuted pm2.62 897. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜓) → ((𝜓 → 𝜑) → 𝜑)) | ||
Theorem | frege45 42903 | Deduce pm2.6 190 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 42904 | 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 pm2.6 190. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((¬ 𝜑 → 𝜓) → ((𝜑 → 𝜓) → 𝜓)) | ||
Theorem | frege47 42905 | 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 42906 | 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 43018. 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 42907 | 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 42908 | Closed form of jaoi 854. 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 42909 | Compare with jaod 856. 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)))) | ||
Here we leverage df-ifp 1061 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 42910 | Justification for ax-frege52a 42911. (Contributed by RP, 17-Apr-2020.) |
⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒))) | ||
Axiom | ax-frege52a 42911 | 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 42912 | 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 214. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) | ||
Theorem | frege53aid 42913 | Specialization of frege53a 42914. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝜑 → ((𝜑 ↔ 𝜓) → 𝜓)) | ||
Theorem | frege53a 42914 | Lemma for frege55a 42922. 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 42915 | Justification for ax-frege54a 42916. Identical to biid 260. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝜑 ↔ 𝜑) | ||
Axiom | ax-frege54a 42916 | 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 260. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) |
⊢ (𝜑 ↔ 𝜑) | ||
Theorem | frege54cor0a 42917 | Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑)) | ||
Theorem | frege54cor1a 42918 | Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ if-(𝜑, 𝜑, ¬ 𝜑) | ||
Theorem | frege55aid 42919 | Lemma for frege57aid 42926. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) | ||
Theorem | frege55lem1a 42920 | 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 42921 | 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 42922 | Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑)) | ||
Theorem | frege55cor1a 42923 | Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)) | ||
Theorem | frege56aid 42924 | Lemma for frege57aid 42926. 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 42925 | 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 42926 | This is the all imporant formula which allows to apply Frege-style definitions and explore their consequences. A closed form of biimpri 227. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)) | ||
Theorem | frege57a 42927 | Analogue of frege57aid 42926. 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 42928 | Identical to anifp 1069. Justification for ax-frege58a 42929. (Contributed by RP, 28-Mar-2020.) |
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) | ||
Axiom | ax-frege58a 42929 | If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2070. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.) |
⊢ ((𝜓 ∧ 𝜒) → if-(𝜑, 𝜓, 𝜒)) | ||
Theorem | frege58acor 42930 | Lemma for frege59a 42931. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.) |
⊢ (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) | ||
Theorem | frege59a 42931 |
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 42867 incorrectly referenced where frege30 42886 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 42932 | Swap antecedents of ax-frege58a 42929. 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 42933 | Lemma for frege65a 42937. 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 42934 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2657 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 42935 | 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 42936 | Lemma for frege65a 42937. 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 42937 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2657 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 42938 | Swap antecedents of frege65a 42937. 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 42939 | Lemma for frege68a 42940. 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 42940 | 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))) | ||
Theorem | axfrege52c 42941 | Justification for ax-frege52c 42942. (Contributed by RP, 24-Dec-2019.) |
⊢ (𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑 → [𝐵 / 𝑥]𝜑)) | ||
Axiom | ax-frege52c 42942 | One side of dfsbcq 3780. 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 42943 | 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 42944 | Lemma for frege102 (via frege92 43009). 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 42945 | Reflexive equality of classes. Identical to eqid 2731. Justification for ax-frege54c 42946. (Contributed by RP, 24-Dec-2019.) |
⊢ 𝐴 = 𝐴 | ||
Axiom | ax-frege54c 42946 | 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 42947 | 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 42948 | Reflexive equality. (Contributed by RP, 24-Dec-2019.) |
⊢ [𝑥 / 𝑦]𝑦 = 𝑥 | ||
Theorem | frege55lem1b 42949* | Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → [𝑥 / 𝑦]𝑦 = 𝑧) → (𝜑 → 𝑥 = 𝑧)) | ||
Theorem | frege55lem2b 42950 | Lemma for frege55b 42951. 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 42951 |
Lemma for frege57b 42953. Proposition 55 of [Frege1879] p. 50.
Note that eqtr2 2755 incorporates eqcom 2738 which is stronger than this proposition which is identical to equcomi 2019. 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 42952 | Lemma for frege57b 42953. 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 42953 | Analogue of frege57aid 42926. 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 42954 | If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2070. Justification for ax-frege58b 42955. (Contributed by RP, 28-Mar-2020.) |
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | ||
Axiom | ax-frege58b 42955 | If ∀𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2070. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.) |
⊢ (∀𝑥𝜑 → [𝑦 / 𝑥]𝜑) | ||
Theorem | frege58bid 42956 | If ∀𝑥𝜑 is affirmed, 𝜑 cannot be denied. Identical to sp 2175. See ax-frege58b 42955 and frege58c 42975 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 42957 | Lemma for frege59b 42958. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (∀𝑥(𝜑 → 𝜓) → ([𝑦 / 𝑥]𝜑 → [𝑦 / 𝑥]𝜓)) | ||
Theorem | frege59b 42958 |
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 42867 incorrectly referenced where frege30 42886 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 42959 | Swap antecedents of ax-frege58b 42955. 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 42960 | Lemma for frege65b 42964. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (([𝑥 / 𝑦]𝜑 → 𝜓) → (∀𝑦𝜑 → 𝜓)) | ||
Theorem | frege62b 42961 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2657 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 42962 | Lemma for frege91 43008. 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 42963 | Lemma for frege65b 42964. 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 42964 |
A kind of Aristotelian inference. This judgement replaces the mode of
inference barbara 2657 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 42965 | Swap antecedents of frege65b 42964. 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 42966 | Lemma for frege68b 42967. 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 42967 | 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)) | ||
Begriffsschrift Chapter II with equivalence of classes (where they are sets). | ||
Theorem | frege53c 42968 | 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 42969* | Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.) |
⊢ 𝐴 ∈ 𝐶 ⇒ ⊢ [𝐴 / 𝑥]𝑥 = 𝐴 | ||
Theorem | frege55lem1c 42970* | Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) |
⊢ ((𝜑 → [𝐴 / 𝑥]𝑥 = 𝐵) → (𝜑 → 𝐴 = 𝐵)) | ||
Theorem | frege55lem2c 42971* | 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 42972 | Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ (𝑥 = 𝐴 → 𝐴 = 𝑥) | ||
Theorem | frege56c 42973* | Lemma for frege57c 42974. 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 42974* | Swap order of implication in ax-frege52c 42942. 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 42975 | Principle related to sp 2175. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.) |
⊢ 𝐴 ∈ 𝐵 ⇒ ⊢ (∀𝑥𝜑 → [𝐴 / 𝑥]𝜑) | ||
Theorem | frege59c 42976 |
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 42867 incorrectly referenced where frege30 42886 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 42977 | Swap antecedents of frege58c 42975. 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 42978 | Lemma for frege65c 42982. 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 42979 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2657 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 42980 | Analogue of frege63b 42962. 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 42981 | Lemma for frege65c 42982. 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 42982 | A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2657 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 42983 | Swap antecedents of frege65c 42982. 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 42984 | Lemma for frege68c 42985. 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 42985 | 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)) | ||
(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 43014 and frege109 43026. dffrege69 42986 through frege75 42992 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 42986* | 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 42987* | Lemma for frege72 42989. 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 42988* | Lemma for frege72 42989. 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 42989 | 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 42990 | Lemma for frege87 43004. 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 42991 | 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 42992* | 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) | ||
𝑝(t+‘𝑅)𝑐 means 𝑐 follows 𝑝 in the 𝑅-sequence. dffrege76 42993 through frege98 43015 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 42993* |
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 42994* | 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 42995* | Commuted form of frege77 42994. 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 42996* | Distributed form of frege78 42995. 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 42997* | Add additional condition to both clauses of frege79 42996. 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 42998 | 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 42999 | Closed-form deduction based on frege81 42998. 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 43000 | Apply commuted form of frege81 42998 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)))) |
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