P. 446 of Theorem List - Metamath Proof Explorer

Theorem List for Metamath Proof Explorer - 44501-44600   *Has distinct variable group(s)

Type	Label	Description
Statement
Theorem	brfvrtrcld 44501*	If two elements are connected by the reflexive-transitive closure of a relation, then they are connected via 𝑛 instances the relation, for some natural number 𝑛. Similar of dfrtrclrec2 15121. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → (𝐴(t*‘𝑅)𝐵 ↔ ∃𝑛 ∈ ℕ0 𝐴(𝑅↑𝑟𝑛)𝐵))
Theorem	fvrtrcllb0d 44502	A restriction of the identity relation is a subset of the reflexive-transitive closure of a set. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ( I ↾ (dom 𝑅 ∪ ran 𝑅)) ⊆ (t*‘𝑅))
Theorem	fvrtrcllb0da 44503	A restriction of the identity relation is a subset of the reflexive-transitive closure of a relation. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → Rel 𝑅)    &   ⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → ( I ↾ ∪ ∪ 𝑅) ⊆ (t*‘𝑅))
Theorem	fvrtrcllb1d 44504	A set is a subset of its image under the reflexive-transitive closure. (Contributed by RP, 22-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    ⇒   ⊢ (𝜑 → 𝑅 ⊆ (t*‘𝑅))
Theorem	dfrtrcl4 44505	Reflexive-transitive closure of a relation, expressed as the union of the zeroth power and the transitive closure. (Contributed by RP, 5-Jun-2020.)
⊢ t* = (𝑟 ∈ V ↦ ((𝑟↑𝑟0) ∪ (t+‘𝑟)))
Theorem	corcltrcl 44506	The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 17-Jun-2020.)
⊢ (r* ∘ t+) = t*
Theorem	cortrcltrcl 44507	Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (t* ∘ t+) = t*
Theorem	corclrtrcl 44508	Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (r* ∘ t*) = t*
Theorem	cotrclrcl 44509	The composition of the reflexive and transitive closures is the reflexive-transitive closure. (Contributed by RP, 21-Jun-2020.)
⊢ (t+ ∘ r*) = t*
Theorem	cortrclrcl 44510	Composition with the reflexive-transitive closure absorbs the reflexive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (t* ∘ r*) = t*
Theorem	cotrclrtrcl 44511	Composition with the reflexive-transitive closure absorbs the transitive closure. (Contributed by RP, 13-Jun-2020.)
⊢ (t+ ∘ t*) = t*
Theorem	cortrclrtrcl 44512	The reflexive-transitive closure is idempotent. (Contributed by RP, 13-Jun-2020.)
⊢ (t* ∘ t*) = t*
21.37.5.5  Adapted from Frege Theorems inspired by Begriffsschrift without restricting form and content to closely parallel those in [Frege1879].
Theorem	frege77d 44513	If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 77 of [Frege1879] p. 62. Compare with frege77 44707. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)    &   ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈)    &   ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑈)
Theorem	frege81d 44514	If the image of 𝑈 is a subset 𝑈, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 81 of [Frege1879] p. 63. Compare with frege81 44711. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)    &   ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑈)
Theorem	frege83d 44515	If the image of the union of 𝑈 and 𝑉 is a subset of the union of 𝑈 and 𝑉, 𝐴 is an element of 𝑈 and 𝐵 follows 𝐴 in the transitive closure of 𝑅, then 𝐵 is an element of the union of 𝑈 and 𝑉. Similar to Proposition 83 of [Frege1879] p. 65. Compare with frege83 44713. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ 𝑈)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)    &   ⊢ (𝜑 → (𝑅 “ (𝑈 ∪ 𝑉)) ⊆ (𝑈 ∪ 𝑉))    ⇒   ⊢ (𝜑 → 𝐵 ∈ (𝑈 ∪ 𝑉))
Theorem	frege96d 44516	If 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 96 of [Frege1879] p. 71. Compare with frege96 44726. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐶 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶)    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
Theorem	frege87d 44517	If the images of both {𝐴} and 𝑈 are subsets of 𝑈 and 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 follows 𝐶 in 𝑅, then 𝐵 is an element of 𝑈. Similar to Proposition 87 of [Frege1879] p. 66. Compare with frege87 44717. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐶 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶)    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    &   ⊢ (𝜑 → (𝑅 “ {𝐴}) ⊆ 𝑈)    &   ⊢ (𝜑 → (𝑅 “ 𝑈) ⊆ 𝑈)    ⇒   ⊢ (𝜑 → 𝐵 ∈ 𝑈)
Theorem	frege91d 44518	If 𝐵 follows 𝐴 in 𝑅 then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 91 of [Frege1879] p. 68. Comparw with frege91 44721. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
Theorem	frege97d 44519	If 𝐴 contains all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 97 of [Frege1879] p. 71. Compare with frege97 44727. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 = ((t+‘𝑅) “ 𝑈))    ⇒   ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)
Theorem	frege98d 44520	If 𝐶 follows 𝐴 and 𝐵 follows 𝐶 in the transitive closure of 𝑅, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 98 of [Frege1879] p. 71. Compare with frege98 44728. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐶 ∈ V)    &   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐶)    &   ⊢ (𝜑 → 𝐶(t+‘𝑅)𝐵)    ⇒   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
Theorem	frege102d 44521	If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 102 of [Frege1879] p. 72. Compare with frege102 44732. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐶 ∈ V)    &   ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    ⇒   ⊢ (𝜑 → 𝐴(t+‘𝑅)𝐵)
Theorem	frege106d 44522	If 𝐵 follows 𝐴 in 𝑅, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in 𝑅. Similar to Proposition 106 of [Frege1879] p. 73. Compare with frege106 44736. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝐴𝑅𝐵)    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))
Theorem	frege108d 44523	If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝑅. Similar to Proposition 108 of [Frege1879] p. 74. Compare with frege108 44738. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐶 ∈ V)    &   ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    ⇒   ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵))
Theorem	frege109d 44524	If 𝐴 contains all elements of 𝑈 and all elements after those in 𝑈 in the transitive closure of 𝑅, then the image under 𝑅 of 𝐴 is a subclass of 𝐴. Similar to Proposition 109 of [Frege1879] p. 74. Compare with frege109 44739. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((t+‘𝑅) “ 𝑈)))    ⇒   ⊢ (𝜑 → (𝑅 “ 𝐴) ⊆ 𝐴)
Theorem	frege114d 44525	If either 𝑅 relates 𝐴 and 𝐵 or 𝐴 and 𝐵 are the same, then either 𝐴 and 𝐵 are the same, 𝑅 relates 𝐴 and 𝐵, 𝑅 relates 𝐵 and 𝐴. Similar to Proposition 114 of [Frege1879] p. 76. Compare with frege114 44744. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵))    ⇒   ⊢ (𝜑 → (𝐴𝑅𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵𝑅𝐴))
Theorem	frege111d 44526	If either 𝐴 and 𝐶 are the same or 𝐶 follows 𝐴 in the transitive closure of 𝑅 and 𝐵 is the successor to 𝐶, then either 𝐴 and 𝐵 are the same or 𝐴 follows 𝐵 or 𝐵 and 𝐴 in the transitive closure of 𝑅. Similar to Proposition 111 of [Frege1879] p. 75. Compare with frege111 44741. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝑅 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ V)    &   ⊢ (𝜑 → 𝐵 ∈ V)    &   ⊢ (𝜑 → 𝐶 ∈ V)    &   ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐶 ∨ 𝐴 = 𝐶))    &   ⊢ (𝜑 → 𝐶𝑅𝐵)    ⇒   ⊢ (𝜑 → (𝐴(t+‘𝑅)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝑅)𝐴))
Theorem	frege122d 44527	If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 is the successor of 𝑋, then 𝐴 and 𝐵 are the same (or 𝐵 follows 𝐴 in the transitive closure of 𝐹). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 44752. (Contributed by RP, 15-Jul-2020.)
⊢ (𝜑 → 𝐴 = (𝐹‘𝑋))    &   ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋))    ⇒   ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵))
Theorem	frege124d 44528	If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then 𝐴 and 𝐵 are the same or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 124 of [Frege1879] p. 80. Compare with frege124 44754. (Contributed by RP, 16-Jul-2020.)
⊢ (𝜑 → 𝐹 ∈ V)    &   ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)    &   ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋))    &   ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵)    &   ⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵))
Theorem	frege126d 44529	If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 follows 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹. Similar to Proposition 126 of [Frege1879] p. 81. Compare with frege126 44756. (Contributed by RP, 16-Jul-2020.)
⊢ (𝜑 → 𝐹 ∈ V)    &   ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)    &   ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋))    &   ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵)    &   ⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴))
Theorem	frege129d 44530	If 𝐹 is a function and (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹, the successor of 𝐴 is either 𝐵 or it follows 𝐵 or it comes before 𝐵 in the transitive closure of 𝐹. Similar to Proposition 129 of [Frege1879] p. 83. Comparw with frege129 44759. (Contributed by RP, 16-Jul-2020.)
⊢ (𝜑 → 𝐹 ∈ V)    &   ⊢ (𝜑 → 𝐴 ∈ dom 𝐹)    &   ⊢ (𝜑 → 𝐶 = (𝐹‘𝐴))    &   ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴))    &   ⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → (𝐵(t+‘𝐹)𝐶 ∨ 𝐵 = 𝐶 ∨ 𝐶(t+‘𝐹)𝐵))
Theorem	frege131d 44531	If 𝐹 is a function and 𝐴 contains all elements of 𝑈 and all elements before or after those elements of 𝑈 in the transitive closure of 𝐹, then the image under 𝐹 of 𝐴 is a subclass of 𝐴. Similar to Proposition 131 of [Frege1879] p. 85. Compare with frege131 44761. (Contributed by RP, 17-Jul-2020.)
⊢ (𝜑 → 𝐹 ∈ V)    &   ⊢ (𝜑 → 𝐴 = (𝑈 ∪ ((◡(t+‘𝐹) “ 𝑈) ∪ ((t+‘𝐹) “ 𝑈))))    &   ⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → (𝐹 “ 𝐴) ⊆ 𝐴)
Theorem	frege133d 44532	If 𝐹 is a function and 𝐴 and 𝐵 both follow 𝑋 in the transitive closure of 𝐹, then (for distinct 𝐴 and 𝐵) either 𝐴 follows 𝐵 or 𝐵 follows 𝐴 in the transitive closure of 𝐹 (or both if it loops). Similar to Proposition 133 of [Frege1879] p. 86. Compare with frege133 44763. (Contributed by RP, 18-Jul-2020.)
⊢ (𝜑 → 𝐹 ∈ V)    &   ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐴)    &   ⊢ (𝜑 → 𝑋(t+‘𝐹)𝐵)    &   ⊢ (𝜑 → Fun 𝐹)    ⇒   ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵(t+‘𝐹)𝐴))
21.37.6  Propositions from _Begriffsschrift_ In 1879, Frege introduced notation for documenting formal reasoning about propositions (and classes) which covered elements of propositional logic, predicate calculus and reasoning about relations. However, due to the pitfalls of naive set theory, adapting this work for inclusion in set.mm required dividing statements about propositions from those about classes and identifying when a restriction to sets is required. For an overview comparing the details of Frege's two-dimensional notation and that used in set.mm, see mmfrege.html. See ru 3746 for discussion of an example of a class that is not a set. Numbered propositions from [Frege1879]. ax-frege1 44557, ax-frege2 44558, ax-frege8 44576, ax-frege28 44597, ax-frege31 44601, ax-frege41 44612, frege52 (see ax-frege52a 44624, frege52b 44656, and ax-frege52c 44655 for translations), frege54 (see ax-frege54a 44629, frege54b 44660 and ax-frege54c 44659 for translations) and frege58 (see ax-frege58a 44642, ax-frege58b 44668 and frege58c 44688 for translations) are considered "core" or axioms. However, at least ax-frege8 44576 can be derived from ax-frege1 44557 and ax-frege2 44558, see axfrege8 44574. Frege introduced implication, negation and the universal quantifier as primitives and did not in the numbered propositions use other logical connectives other than equivalence introduced in ax-frege52a 44624, frege52b 44656, and ax-frege52c 44655. In dffrege69 44699, Frege introduced 𝑅 hereditary 𝐴 to say that relation 𝑅, when restricted to operate on elements of class 𝐴, will only have elements of class 𝐴 in its domain; see df-he 44540 for a definition in terms of image and subset. In dffrege76 44706, Frege introduced notation for the concept of two sets related by the transitive closure of a relation, for which we write 𝑋(t+‘𝑅)𝑌, which requires 𝑅 to also be a set. In dffrege99 44729, Frege introduced notation for the concept of two sets either identical or related by the transitive closure of a relation, for which we write 𝑋((t+‘𝑅) ∪ I )𝑌, which is a superclass of sets related by the reflexive-transitive relation 𝑋(t*‘𝑅)𝑌. Finally, in dffrege115 44745, Frege introduced notation for the concept of a relation having the property elements in its domain pair up with only one element each in its range, for which we write Fun ◡◡𝑅 (to ignore any non-relational content of the class 𝑅). Frege did this without the expressing concept of a relation (or its transitive closure) as a class, and needed to invent conventions for discussing indeterminate propositions with two slots free and how to recognize which of the slots was domain and which was range. See mmfrege.html 44745 for details. English translations for specific propositions lifted in part from a translation by Stefan Bauer-Mengelberg as reprinted in From Frege to Goedel: A Source Book in Mathematical Logic, 1879-1931. An attempt to align these propositions in the larger set.mm database has also been made. See frege77d 44513 for an example.
21.37.6.1  _Begriffsschrift_ Chapter I Section 2 introduces the turnstile ⊢ which turns an idea which may be true 𝜑 into an assertion that it does hold true ⊢ 𝜑. Section 5 introduces implication, (𝜑 → 𝜓). Section 6 introduces the single rule of interference relied upon, modus ponens ax-mp 5. Section 7 introduces negation and with in synonyms for or (¬ 𝜑 → 𝜓) , and ¬ (𝜑 → ¬ 𝜓), and two for exclusive-or corresponding to df-or 862, df-an 402, dfxor4 44533, dfxor5 44534. Section 8 introduces the problematic notation for identity of conceptual content which must be separated into cases for biconditional (𝜑 ↔ 𝜓) or class equality 𝐴 = 𝐵 in this adaptation. Section 10 introduces "truth functions" for one or two variables in equally troubling notation, as the arguments may be understood to be logical predicates or collections. Here f(𝜑) is interpreted to mean if-(𝜑, 𝜓, 𝜒) where the content of the "function" is specified by the latter two arguments or logical equivalent, while g(𝐴) is read as 𝐴 ∈ 𝐺 and h(𝐴, 𝐵) as 𝐴𝐻𝐵. This necessarily introduces a need for set theory as both 𝐴 ∈ 𝐺 and 𝐴𝐻𝐵 cannot hold unless 𝐴 is a set. (Also 𝐵.) Section 11 introduces notation for generality, but there is no standard notation for generality when the variable is a proposition because it was realized after Frege that the universe of all possible propositions includes paradoxical constructions leading to the failure of naive set theory. So adopting f(𝜑) as if-(𝜑, 𝜓, 𝜒) would result in the translation of ∀𝜑 f (𝜑) as (𝜓 ∧ 𝜒). For collections, we must generalize over set variables or run into the same problems; this leads to ∀𝐴 g(𝐴) being translated as ∀𝑎𝑎 ∈ 𝐺 and so forth. Under this interpreation the text of section 11 gives us sp 2222 (or simpl 488 and simpr 490 and anifp 1088 in the propositional case) and statements similar to cbvalivw 2040, ax-gen 1828, alrimiv 1960, and alrimdv 1962. These last four introduce a generality and have no useful definition in terms of propositional variables. Section 12 introduces some combinations of primitive symbols and their human language counterparts. Using class notation, these can also be expressed without dummy variables. All are A, ∀𝑥𝑥 ∈ 𝐴, ¬ ∃𝑥¬ 𝑥 ∈ 𝐴 alex 1859, 𝐴 = V eqv 3468; Some are not B, ¬ ∀𝑥𝑥 ∈ 𝐵, ∃𝑥¬ 𝑥 ∈ 𝐵 exnal 1860, 𝐵 ⊊ V pssv 4372, 𝐵 ≠ V nev 44537; There are no C, ∀𝑥¬ 𝑥 ∈ 𝐶, ¬ ∃𝑥𝑥 ∈ 𝐶 alnex 1814, 𝐶 = ∅ eq0 4307; There exist D, ¬ ∀𝑥¬ 𝑥 ∈ 𝐷, ∃𝑥𝑥 ∈ 𝐷 df-ex 1813, ∅ ⊊ 𝐷 0pss 4370, 𝐷 ≠ ∅ n0 4310. Notation for relations between expressions also can be written in various ways. All E are P, ∀𝑥(𝑥 ∈ 𝐸 → 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐸 ∧ ¬ 𝑥 ∈ 𝑃) dfss6 3930, 𝐸 = (𝐸 ∩ 𝑃) dfss2 3926, 𝐸 ⊆ 𝑃 df-ss 3925; No F are P, ∀𝑥(𝑥 ∈ 𝐹 → ¬ 𝑥 ∈ 𝑃), ¬ ∃𝑥(𝑥 ∈ 𝐹 ∧ 𝑥 ∈ 𝑃) alinexa 1876, (𝐹 ∩ 𝑃) = ∅ disj1 4415; Some G are not P, ¬ ∀𝑥(𝑥 ∈ 𝐺 → 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐺 ∧ ¬ 𝑥 ∈ 𝑃) exanali 1892, (𝐺 ∩ 𝑃) ⊊ 𝐺 nssinpss 4223, ¬ 𝐺 ⊆ 𝑃 nss 4004; Some H are P, ¬ ∀𝑥(𝑥 ∈ 𝐻 → ¬ 𝑥 ∈ 𝑃), ∃𝑥(𝑥 ∈ 𝐻 ∧ 𝑥 ∈ 𝑃) exnalimn 1877, ∅ ⊊ (𝐻 ∩ 𝑃) 0pssin 44538, (𝐻 ∩ 𝑃) ≠ ∅ ndisj 4328.
Theorem	dfxor4 44533	Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((¬ 𝜑 → 𝜓) → ¬ (𝜑 → ¬ 𝜓)))
Theorem	dfxor5 44534	Express exclusive-or in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 14-Apr-2020.)
⊢ ((𝜑 ⊻ 𝜓) ↔ ¬ ((𝜑 → ¬ 𝜓) → ¬ (¬ 𝜑 → 𝜓)))
Theorem	df3or2 44535	Express triple-or in terms of implication and negation. Statement in [Frege1879] p. 11. (Contributed by RP, 25-Jul-2020.)
⊢ ((𝜑 ∨ 𝜓 ∨ 𝜒) ↔ (¬ 𝜑 → (¬ 𝜓 → 𝜒)))
Theorem	df3an2 44536	Express triple-and in terms of implication and negation. Statement in [Frege1879] p. 12. (Contributed by RP, 25-Jul-2020.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) ↔ ¬ (𝜑 → (𝜓 → ¬ 𝜒)))
Theorem	nev 44537*	Express that not every set is in a class. (Contributed by RP, 16-Apr-2020.)
⊢ (𝐴 ≠ V ↔ ¬ ∀𝑥 𝑥 ∈ 𝐴)
Theorem	0pssin 44538*	Express that an intersection is not empty. (Contributed by RP, 16-Apr-2020.)
⊢ (∅ ⊊ (𝐴 ∩ 𝐵) ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐵))
21.37.6.2  _Begriffsschrift_ Notation hints The statement 𝑅 hereditary 𝐴 means relation 𝑅 is hereditary (in the sense of Frege) in the class 𝐴 or (𝑅 “ 𝐴) ⊆ 𝐴. The former is only a slight reduction in the number of symbols, but this reduces the number of floating hypotheses needed to be checked. As Frege was not using the language of classes or sets, this naturally 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.
Syntax	whe 44539	The property of relation 𝑅 being hereditary in class 𝐴.
wff 𝑅 hereditary 𝐴
Definition	df-he 44540	The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 “ 𝐴) ⊆ 𝐴)
Theorem	dfhe2 44541	The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
⊢ (𝑅 hereditary 𝐴 ↔ (𝑅 ↾ 𝐴) ⊆ (𝐴 × 𝐴))
Theorem	dfhe3 44542*	The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
⊢ (𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 → ∀𝑦(𝑥𝑅𝑦 → 𝑦 ∈ 𝐴)))
Theorem	heeq12 44543	Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
⊢ ((𝑅 = 𝑆 ∧ 𝐴 = 𝐵) → (𝑅 hereditary 𝐴 ↔ 𝑆 hereditary 𝐵))
Theorem	heeq1 44544	Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
⊢ (𝑅 = 𝑆 → (𝑅 hereditary 𝐴 ↔ 𝑆 hereditary 𝐴))
Theorem	heeq2 44545	Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
⊢ (𝐴 = 𝐵 → (𝑅 hereditary 𝐴 ↔ 𝑅 hereditary 𝐵))
Theorem	sbcheg 44546	Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)
⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶 ↔ ⦋𝐴 / 𝑥⦌𝐵 hereditary ⦋𝐴 / 𝑥⦌𝐶))
Theorem	hess 44547	Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
⊢ (𝑆 ⊆ 𝑅 → (𝑅 hereditary 𝐴 → 𝑆 hereditary 𝐴))
Theorem	xphe 44548	Any Cartesian product is hereditary in its second class. (Contributed by RP, 27-Mar-2020.) (Proof shortened by OpenAI, 3-Jul-2020.)
⊢ (𝐴 × 𝐵) hereditary 𝐵
Theorem	0he 44549	The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
⊢ ∅ hereditary 𝐴
Theorem	0heALT 44550	The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ∅ hereditary 𝐴
Theorem	he0 44551	Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
⊢ 𝐴 hereditary ∅
Theorem	unhe1 44552	The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)
⊢ ((𝑅 hereditary 𝐴 ∧ 𝑆 hereditary 𝐴) → (𝑅 ∪ 𝑆) hereditary 𝐴)
Theorem	snhesn 44553	Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
⊢ {⟨𝐴, 𝐴⟩} hereditary {𝐵}
Theorem	idhe 44554	The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
⊢ I hereditary 𝐴
Theorem	psshepw 44555	The relation between sets and their proper subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
⊢ ◡ [⊊] hereditary 𝒫 𝐴
Theorem	sshepw 44556	The relation between sets and their subsets is hereditary in the powerclass of any class. (Contributed by RP, 28-Mar-2020.)
⊢ (◡ [⊊] ∪ I ) hereditary 𝒫 𝐴
21.37.6.3  _Begriffsschrift_ Chapter II Implication
Axiom	ax-frege1 44557	The case in which 𝜑 is denied, 𝜓 is affirmed, and 𝜑 is affirmed is excluded. This is evident since 𝜑 cannot at the same time be denied and affirmed. Axiom 1 of [Frege1879] p. 26. Identical to ax-1 6. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜑))
Axiom	ax-frege2 44558	If a proposition 𝜒 is a necessary consequence of two propositions 𝜓 and 𝜑 and one of those, 𝜓, is in turn a necessary consequence of the other, 𝜑, then the proposition 𝜒 is a necessary consequence of the latter one, 𝜑, alone. Axiom 2 of [Frege1879] p. 26. Identical to ax-2 7. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜑 → 𝜓) → (𝜑 → 𝜒)))
Theorem	rp-simp2-frege 44559	Simplification of triple conjunction. Compare with simp2 1155. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → (𝜒 → 𝜓)))
Theorem	rp-simp2 44560	Simplification of triple conjunction. Identical to simp2 1155. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 ∧ 𝜓 ∧ 𝜒) → 𝜓)
Theorem	rp-frege3g 44561	Add antecedent to ax-frege2 44558. More general statement than frege3 44562. Like ax-frege2 44558, it is essentially a closed form of mpd 16, however it has an extra antecedent. It would be more natural to prove from a1i 11 and ax-frege2 44558 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → ((𝜓 → (𝜒 → 𝜃)) → ((𝜓 → 𝜒) → (𝜓 → 𝜃))))
Theorem	frege3 44562	Add antecedent to ax-frege2 44558. Special case of rp-frege3g 44561. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜑 → 𝜓)) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))))
Theorem	rp-misc1-frege 44563	Double-use of ax-frege2 44558. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜓)) → ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → 𝜒)))
Theorem	rp-frege24 44564	Introducing an embedded antecedent. Alternate proof for frege24 44582. Closed form for a1d 26. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))
Theorem	rp-frege4g 44565	Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → ((𝜓 → 𝜒) → (𝜓 → 𝜃))))
Theorem	frege4 44566	Special case of closed form of a2d 30. Special case of rp-frege4g 44565. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((𝜑 → 𝜓) → (𝜒 → (𝜑 → 𝜓))) → ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓))))
Theorem	frege5 44567	A closed form of syl 18. Identical to imim2 59. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → 𝜑) → (𝜒 → 𝜓)))
Theorem	rp-7frege 44568	Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜃 → ((𝜑 → 𝜓) → (𝜑 → 𝜒))))
Theorem	rp-4frege 44569	Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → ((𝜓 → 𝜑) → 𝜒)) → (𝜑 → 𝜒))
Theorem	rp-6frege 44570	Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
⊢ (𝜑 → ((𝜓 → ((𝜒 → 𝜓) → 𝜃)) → (𝜓 → 𝜃)))
Theorem	rp-8frege 44571	Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → (𝜓 → ((𝜒 → 𝜓) → 𝜃))) → (𝜑 → (𝜓 → 𝜃)))
Theorem	rp-frege25 44572	Closed form for a1dd 51. Alternate route to Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))
Theorem	frege6 44573	A closed form of imim2d 58 which is a deduction adding nested antecedents. Proposition 6 of [Frege1879] p. 33. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → ((𝜃 → 𝜓) → (𝜃 → 𝜒))))
Theorem	axfrege8 44574	Swap antecedents. Identical to pm2.04 91. This demonstrates that Axiom 8 of [Frege1879] p. 35 is redundant. Proof follows closely proof of pm2.04 91 in https://us.metamath.org/mmsolitaire/pmproofs.txt 91, but in the style of Frege's 1879 work. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
Theorem	frege7 44575	A closed form of syl6 36. The first antecedent is used to replace the consequent of the second antecedent. Proposition 7 of [Frege1879] p. 34. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜒 → (𝜃 → 𝜑)) → (𝜒 → (𝜃 → 𝜓))))
Axiom	ax-frege8 44576	Swap antecedents. If two conditions have a proposition as a consequence, their order is immaterial. Third axiom of Frege's 1879 work but identical to pm2.04 91 which can be proved from only ax-mp 5, ax-frege1 44557, and ax-frege2 44558. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (𝜑 → 𝜒)))
Theorem	frege26 44577	Identical to idd 25. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → (𝜓 → 𝜓))
Theorem	frege27 44578	We cannot (at the same time) affirm 𝜑 and deny 𝜑. Identical to id 23. Proposition 27 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (𝜑 → 𝜑)
Theorem	frege9 44579	Closed form of syl 18 with swapped antecedents. This proposition differs from frege5 44567 only in an unessential way. Identical to imim1 84. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → ((𝜓 → 𝜒) → (𝜑 → 𝜒)))
Theorem	frege12 44580	A closed form of com23 87. Proposition 12 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜑 → (𝜒 → (𝜓 → 𝜃))))
Theorem	frege11 44581	Elimination of a nested antecedent as a partial converse of ja 188. If the proposition that 𝜓 takes place or 𝜑 does not is a sufficient condition for 𝜒, then 𝜓 by itself is a sufficient condition for 𝜒. Identical to jarr 107. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → (𝜓 → 𝜒))
Theorem	frege24 44582	Closed form for a1d 26. Deduction introducing an embedded antecedent. Identical to rp-frege24 44564 which was proved without relying on ax-frege8 44576. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → 𝜓) → (𝜑 → (𝜒 → 𝜓)))
Theorem	frege16 44583	A closed form of com34 92. Proposition 16 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) → (𝜑 → (𝜓 → (𝜃 → (𝜒 → 𝜏)))))
Theorem	frege25 44584	Closed form for a1dd 51. Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (𝜓 → (𝜃 → 𝜒))))
Theorem	frege18 44585	Closed form of a syllogism followed by a swap of antecedents. Proposition 18 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜃 → 𝜑) → (𝜓 → (𝜃 → 𝜒))))
Theorem	frege22 44586	A closed form of com45 98. Proposition 22 of [Frege1879] p. 41. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏 → 𝜂))))) → (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃 → 𝜂))))))
Theorem	frege10 44587	Result commuting antecedents within an antecedent. Proposition 10 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((𝜑 → (𝜓 → 𝜒)) → 𝜃) → ((𝜓 → (𝜑 → 𝜒)) → 𝜃))
Theorem	frege17 44588	A closed form of com3l 90. Proposition 17 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜓 → (𝜒 → (𝜑 → 𝜃))))
Theorem	frege13 44589	A closed form of com3r 88. Proposition 13 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → (𝜒 → (𝜑 → (𝜓 → 𝜃))))
Theorem	frege14 44590	Closed form of a deduction based on com3r 88. Proposition 14 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) → (𝜑 → (𝜃 → (𝜓 → (𝜒 → 𝜏)))))
Theorem	frege19 44591	A closed form of syl6 36. Proposition 19 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → ((𝜒 → 𝜃) → (𝜑 → (𝜓 → 𝜃))))
Theorem	frege23 44592	Syllogism followed by rotation of three antecedents. Proposition 23 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜒 → 𝜃))) → ((𝜏 → 𝜑) → (𝜓 → (𝜒 → (𝜏 → 𝜃)))))
Theorem	frege15 44593	A closed form of com4r 95. Proposition 15 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → (𝜒 → (𝜃 → 𝜏)))) → (𝜃 → (𝜑 → (𝜓 → (𝜒 → 𝜏)))))
Theorem	frege21 44594	Replace antecedent in antecedent. Proposition 21 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ (((𝜑 → 𝜓) → 𝜒) → ((𝜑 → 𝜃) → ((𝜃 → 𝜓) → 𝜒)))
Theorem	frege20 44595	A closed form of syl8 77. 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)))))
21.37.6.4  _Begriffsschrift_ Chapter II Implication and Negation
Theorem	axfrege28 44596	Contraposition. Identical to con3 154. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.)
⊢ ((𝜑 → 𝜓) → (¬ 𝜓 → ¬ 𝜑))
Axiom	ax-frege28 44597	Contraposition. Identical to con3 154. 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 44598	Closed form of con3d 153. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
Theorem	frege30 44599	Commuted, closed form of con3d 153. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
⊢ ((𝜑 → (𝜓 → 𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))
Theorem	axfrege31 44600	Identical to notnotr 131. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
⊢ (¬ ¬ 𝜑 → 𝜑)