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Theorem List for Metamath Proof Explorer - 40001-40100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdfhe3 40001* The property of relation 𝑅 being hereditary in class 𝐴. (Contributed by RP, 27-Mar-2020.)
(𝑅 hereditary 𝐴 ↔ ∀𝑥(𝑥𝐴 → ∀𝑦(𝑥𝑅𝑦𝑦𝐴)))
 
Theoremheeq12 40002 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
((𝑅 = 𝑆𝐴 = 𝐵) → (𝑅 hereditary 𝐴𝑆 hereditary 𝐵))
 
Theoremheeq1 40003 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
(𝑅 = 𝑆 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
 
Theoremheeq2 40004 Equality law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
(𝐴 = 𝐵 → (𝑅 hereditary 𝐴𝑅 hereditary 𝐵))
 
Theoremsbcheg 40005 Distribute proper substitution through herditary relation. (Contributed by RP, 29-Jun-2020.)
(𝐴𝑉 → ([𝐴 / 𝑥]𝐵 hereditary 𝐶𝐴 / 𝑥𝐵 hereditary 𝐴 / 𝑥𝐶))
 
Theoremhess 40006 Subclass law for relations being herditary over a class. (Contributed by RP, 27-Mar-2020.)
(𝑆𝑅 → (𝑅 hereditary 𝐴𝑆 hereditary 𝐴))
 
Theoremxphe 40007 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
 
Theorem0he 40008 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.)
∅ hereditary 𝐴
 
Theorem0heALT 40009 The empty relation is hereditary in any class. (Contributed by RP, 27-Mar-2020.) (New usage is discouraged.) (Proof modification is discouraged.)
∅ hereditary 𝐴
 
Theoremhe0 40010 Any relation is hereditary in the empty set. (Contributed by RP, 27-Mar-2020.)
𝐴 hereditary ∅
 
Theoremunhe1 40011 The union of two relations hereditary in a class is also hereditary in a class. (Contributed by RP, 28-Mar-2020.)
((𝑅 hereditary 𝐴𝑆 hereditary 𝐴) → (𝑅𝑆) hereditary 𝐴)
 
Theoremsnhesn 40012 Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
{⟨𝐴, 𝐴⟩} hereditary {𝐵}
 
Theoremidhe 40013 The identity relation is hereditary in any class. (Contributed by RP, 28-Mar-2020.)
I hereditary 𝐴
 
Theorempsshepw 40014 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
 
Theoremsshepw 40015 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
 
20.30.3.3  _Begriffsschrift_ Chapter II Implication
 
Axiomax-frege1 40016 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 → (𝜓𝜑))
 
Axiomax-frege2 40017 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 → (𝜓𝜒)) → ((𝜑𝜓) → (𝜑𝜒)))
 
Theoremrp-simp2-frege 40018 Simplification of triple conjunction. Compare with simp2 1129. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜓)))
 
Theoremrp-simp2 40019 Simplification of triple conjunction. Identical to simp2 1129. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓𝜒) → 𝜓)
 
Theoremrp-frege3g 40020 Add antecedent to ax-frege2 40017. More general statement than frege3 40021. Like ax-frege2 40017, it is essentially a closed form of mpd 15, however it has an extra antecedent.

It would be more natural to prove from a1i 11 and ax-frege2 40017 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

(𝜑 → ((𝜓 → (𝜒𝜃)) → ((𝜓𝜒) → (𝜓𝜃))))
 
Theoremfrege3 40021 Add antecedent to ax-frege2 40017. Special case of rp-frege3g 40020. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒 → (𝜑𝜓)) → ((𝜒𝜑) → (𝜒𝜓))))
 
Theoremrp-misc1-frege 40022 Double-use of ax-frege2 40017. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑 → (𝜓𝜒)) → (𝜑𝜓)) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
 
Theoremrp-frege24 40023 Introducing an embedded antecedent. Alternate proof for frege24 40041. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))
 
Theoremrp-frege4g 40024 Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → ((𝜓𝜒) → (𝜓𝜃))))
 
Theoremfrege4 40025 Special case of closed form of a2d 29. Special case of rp-frege4g 40024. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (𝜒 → (𝜑𝜓))) → ((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓))))
 
Theoremfrege5 40026 A closed form of syl 17. Identical to imim2 58. Theorem *2.05 of [WhiteheadRussell] p. 100. Proposition 5 of [Frege1879] p. 32. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓)))
 
Theoremrp-7frege 40027 Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓𝜒)) → (𝜃 → ((𝜑𝜓) → (𝜑𝜒))))
 
Theoremrp-4frege 40028 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
((𝜑 → ((𝜓𝜑) → 𝜒)) → (𝜑𝜒))
 
Theoremrp-6frege 40029 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
(𝜑 → ((𝜓 → ((𝜒𝜓) → 𝜃)) → (𝜓𝜃)))
 
Theoremrp-8frege 40030 Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → ((𝜒𝜓) → 𝜃))) → (𝜑 → (𝜓𝜃)))
 
Theoremrp-frege25 40031 Closed form for a1dd 50. Alternate route to Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))
 
Theoremfrege6 40032 A closed form of imim2d 57 which is a deduction adding nested antecedents. Proposition 6 of [Frege1879] p. 33. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → ((𝜃𝜓) → (𝜃𝜒))))
 
Theoremaxfrege8 40033 Swap antecedents. Identical to pm2.04 90. This demonstrates that Axiom 8 of [Frege1879] p. 35 is redundant.

Proof follows closely proof of pm2.04 90 in https://us.metamath.org/mmsolitaire/pmproofs.txt 90, but in the style of Frege's 1879 work. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.) (Proof modification is discouraged.)

((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
 
Theoremfrege7 40034 A closed form of syl6 35. 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.)
((𝜑𝜓) → ((𝜒 → (𝜃𝜑)) → (𝜒 → (𝜃𝜓))))
 
Axiomax-frege8 40035 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 90 which can be proved from only ax-mp 5, ax-frege1 40016, and ax-frege2 40017. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
 
Theoremfrege26 40036 Identical to idd 24. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜓))
 
Theoremfrege27 40037 We cannot (at the same time) affirm 𝜑 and deny 𝜑. Identical to id 22. Proposition 27 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑𝜑)
 
Theoremfrege9 40038 Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 40026 only in an unessential way. Identical to imim1 83. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremfrege12 40039 A closed form of com23 86. Proposition 12 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → (𝜒 → (𝜓𝜃))))
 
Theoremfrege11 40040 Elimination of a nested antecedent as a partial converse of ja 187. 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 106. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 
Theoremfrege24 40041 Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 40023 which was proved without relying on ax-frege8 40035. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))
 
Theoremfrege16 40042 A closed form of com34 91. Proposition 16 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃𝜏)))) → (𝜑 → (𝜓 → (𝜃 → (𝜒𝜏)))))
 
Theoremfrege25 40043 Closed form for a1dd 50. Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))
 
Theoremfrege18 40044 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 → (𝜓𝜒)) → ((𝜃𝜑) → (𝜓 → (𝜃𝜒))))
 
Theoremfrege22 40045 A closed form of com45 97. Proposition 22 of [Frege1879] p. 41. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃 → (𝜏𝜂))))) → (𝜑 → (𝜓 → (𝜒 → (𝜏 → (𝜃𝜂))))))
 
Theoremfrege10 40046 Result commuting antecedents within an antecedent. Proposition 10 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑 → (𝜓𝜒)) → 𝜃) → ((𝜓 → (𝜑𝜒)) → 𝜃))
 
Theoremfrege17 40047 A closed form of com3l 89. Proposition 17 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜓 → (𝜒 → (𝜑𝜃))))
 
Theoremfrege13 40048 A closed form of com3r 87. Proposition 13 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜒 → (𝜑 → (𝜓𝜃))))
 
Theoremfrege14 40049 Closed form of a deduction based on com3r 87. Proposition 14 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃𝜏)))) → (𝜑 → (𝜃 → (𝜓 → (𝜒𝜏)))))
 
Theoremfrege19 40050 A closed form of syl6 35. Proposition 19 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜒𝜃) → (𝜑 → (𝜓𝜃))))
 
Theoremfrege23 40051 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 → (𝜒𝜃))) → ((𝜏𝜑) → (𝜓 → (𝜒 → (𝜏𝜃)))))
 
Theoremfrege15 40052 A closed form of com4r 94. Proposition 15 of [Frege1879] p. 38. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒 → (𝜃𝜏)))) → (𝜃 → (𝜑 → (𝜓 → (𝜒𝜏)))))
 
Theoremfrege21 40053 Replace antecedent in antecedent. Proposition 21 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → 𝜒) → ((𝜑𝜃) → ((𝜃𝜓) → 𝜒)))
 
Theoremfrege20 40054 A closed form of syl8 76. Proposition 20 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → ((𝜃𝜏) → (𝜑 → (𝜓 → (𝜒𝜏)))))
 
20.30.3.4  _Begriffsschrift_ Chapter II Implication and Negation
 
Theoremaxfrege28 40055 Contraposition. Identical to con3 156. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
Axiomax-frege28 40056 Contraposition. Identical to con3 156. Theorem *2.16 of [WhiteheadRussell] p. 103. Axiom 28 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
Theoremfrege29 40057 Closed form of con3d 155. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
 
Theoremfrege30 40058 Commuted, closed form of con3d 155. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))
 
Theoremaxfrege31 40059 Identical to notnotr 132. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
(¬ ¬ 𝜑𝜑)
 
Axiomax-frege31 40060 𝜑 cannot be denied and (at the same time ) ¬ ¬ 𝜑 affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnotr 132. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)
 
Theoremfrege32 40061 Deduce con1 148 from con3 156. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
 
Theoremfrege33 40062 If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 148. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → (¬ 𝜓𝜑))
 
Theoremfrege34 40063 If as a conseqence 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 147. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓)))
 
Theoremfrege35 40064 Commuted, closed form of con1d 147. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (¬ 𝜒 → (𝜑𝜓)))
 
Theoremfrege36 40065 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 → (¬ 𝜑𝜓))
 
Theoremfrege37 40066 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 871. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → 𝜒) → (𝜑𝜒))
 
Theoremfrege38 40067 Identical to pm2.21 123. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝜑 → (𝜑𝜓))
 
Theoremfrege39 40068 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.)
((¬ 𝜑𝜑) → (¬ 𝜑𝜓))
 
Theoremfrege40 40069 Anything implies pm2.18 128. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝜑 → ((¬ 𝜓𝜓) → 𝜓))
 
Theoremaxfrege41 40070 Identical to notnot 144. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)
(𝜑 → ¬ ¬ 𝜑)
 
Axiomax-frege41 40071 The affirmation of 𝜑 denies the denial of 𝜑. Identical to notnot 144. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑 → ¬ ¬ 𝜑)
 
Theoremfrege42 40072 Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
¬ ¬ (𝜑𝜑)
 
Theoremfrege43 40073 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.)
((¬ 𝜑𝜑) → 𝜑)
 
Theoremfrege44 40074 Similar to a commuted pm2.62 893. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜓𝜑) → 𝜑))
 
Theoremfrege45 40075 Deduce pm2.6 192 from con1 148. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → (¬ 𝜓𝜑)) → ((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓)))
 
Theoremfrege46 40076 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 pm2.6 192. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theoremfrege47 40077 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.)
((¬ 𝜑𝜓) → ((𝜓𝜒) → ((𝜑𝜒) → 𝜒)))
 
Theoremfrege48 40078 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 40190. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → ((𝜒𝜃) → ((𝜓𝜃) → (𝜑𝜃))))
 
Theoremfrege49 40079 Closed form of deduction with disjunction. Proposition 49 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜑𝜒) → ((𝜓𝜒) → 𝜒)))
 
Theoremfrege50 40080 Closed form of jaoi 851. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜓) → ((¬ 𝜑𝜒) → 𝜓)))
 
Theoremfrege51 40081 Compare with jaod 853. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) → (𝜑 → ((¬ 𝜓𝜃) → 𝜒))))
 
20.30.3.5  _Begriffsschrift_ Chapter II with logical equivalence

Here we leverage df-ifp 1055 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 ) (𝜑𝜓) with if-(𝜓, 𝜒, 𝜃) to reason about the values of the "function." Likewise, we replace the similarly illegal concept 𝜓𝜑 with (𝜒𝜃).

 
Theoremaxfrege52a 40082 Justification for ax-frege52a 40083. (Contributed by RP, 17-Apr-2020.)
((𝜑𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
 
Axiomax-frege52a 40083 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.)
((𝜑𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
 
Theoremfrege52aid 40084 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 216. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremfrege53aid 40085 Specialization of frege53a 40086. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → ((𝜑𝜓) → 𝜓))
 
Theoremfrege53a 40086 Lemma for frege55a 40094. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(if-(𝜑, 𝜃, 𝜒) → ((𝜑𝜓) → if-(𝜓, 𝜃, 𝜒)))
 
Theoremaxfrege54a 40087 Justification for ax-frege54a 40088. Identical to biid 262. (Contributed by RP, 24-Dec-2019.)
(𝜑𝜑)
 
Axiomax-frege54a 40088 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 262. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑𝜑)
 
Theoremfrege54cor0a 40089 Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜓𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
 
Theoremfrege54cor1a 40090 Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if-(𝜑, 𝜑, ¬ 𝜑)
 
Theoremfrege55aid 40091 Lemma for frege57aid 40098. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremfrege55lem1a 40092 Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜏 → if-(𝜓, 𝜑, ¬ 𝜑)) → (𝜏 → (𝜓𝜑)))
 
Theoremfrege55lem2a 40093 Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))
 
Theoremfrege55a 40094 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))
 
Theoremfrege55cor1a 40095 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremfrege56aid 40096 Lemma for frege57aid 40098. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (𝜑𝜓)) → ((𝜓𝜑) → (𝜑𝜓)))
 
Theoremfrege56a 40097 Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))))
 
Theoremfrege57aid 40098 This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 229. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremfrege57a 40099 Analogue of frege57aid 40098. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))
 
Theoremaxfrege58a 40100 Identical to anifp 1062. Justification for ax-frege58a 40101. (Contributed by RP, 28-Mar-2020.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
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