HomeHome Metamath Proof Explorer
Theorem List (p. 371 of 425)
< Previous  Next >
Bad symbols? Try the
GIF version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26947)
  Hilbert Space Explorer  Hilbert Space Explorer
(26948-28472)
  Users' Mathboxes  Users' Mathboxes
(28473-42426)
 

Theorem List for Metamath Proof Explorer - 37001-37100   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
20.26.3.3  _Begriffsschrift_ Chapter II Implication
 
Axiomax-frege1 37001 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 37002 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 37003 Simplification of triple conjunction. Compare with simp2 1054. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜓)))
 
Theoremrp-simp2 37004 Simplification of triple conjunction. Identical to simp2 1054. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓𝜒) → 𝜓)
 
Theoremrp-frege3g 37005 Add antecedent to ax-frege2 37002. More general statement than frege3 37006. Like ax-frege2 37002, 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 37002 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

(𝜑 → ((𝜓 → (𝜒𝜃)) → ((𝜓𝜒) → (𝜓𝜃))))
 
Theoremfrege3 37006 Add antecedent to ax-frege2 37002. Special case of rp-frege3g 37005. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒 → (𝜑𝜓)) → ((𝜒𝜑) → (𝜒𝜓))))
 
Theoremrp-misc1-frege 37007 Double-use of ax-frege2 37002. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑 → (𝜓𝜒)) → (𝜑𝜓)) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
 
Theoremrp-frege24 37008 Introducing an embedded antecedent. Alternate proof for frege24 37026. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))
 
Theoremrp-frege4g 37009 Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → ((𝜓𝜒) → (𝜓𝜃))))
 
Theoremfrege4 37010 Special case of closed form of a2d 29. Special case of rp-frege4g 37009. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (𝜒 → (𝜑𝜓))) → ((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓))))
 
Theoremfrege5 37011 A closed form of syl 17. Identical to imim2 55. 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 37012 Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓𝜒)) → (𝜃 → ((𝜑𝜓) → (𝜑𝜒))))
 
Theoremrp-4frege 37013 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
((𝜑 → ((𝜓𝜑) → 𝜒)) → (𝜑𝜒))
 
Theoremrp-6frege 37014 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
(𝜑 → ((𝜓 → ((𝜒𝜓) → 𝜃)) → (𝜓𝜃)))
 
Theoremrp-8frege 37015 Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → ((𝜒𝜓) → 𝜃))) → (𝜑 → (𝜓𝜃)))
 
Theoremrp-frege25 37016 Closed form for a1dd 47. Alternate route to Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))
 
Theoremfrege6 37017 A closed form of imim2d 54 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 37018 Swap antecedents. Identical to pm2.04 87. This demonstrates that Axiom 8 of [Frege1879] p. 35 is redundant.

Proof follows closely proof of pm2.04 87 in http://us.metamath.org/mmsolitaire/pmproofs.txt, 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 37019 A closed form of syl6 34. 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 37020 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 87 which can be proved from only ax-mp 5, ax-frege1 37001, and ax-frege2 37002. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
 
Theoremfrege26 37021 Identical to idd 24. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜓))
 
Theoremfrege27 37022 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 37023 Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 37011 only in an unessential way. Identical to imim1 80. Proposition 9 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜓𝜒) → (𝜑𝜒)))
 
Theoremfrege12 37024 A closed form of com23 83. Proposition 12 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → (𝜒 → (𝜓𝜃))))
 
Theoremfrege11 37025 Elimination of a nested antecedent as a partial converse of ja 171. 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 103. Proposition 11 of [Frege1879] p. 36. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → 𝜒) → (𝜓𝜒))
 
Theoremfrege24 37026 Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 37008 which was proved without relying on ax-frege8 37020. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))
 
Theoremfrege16 37027 A closed form of com34 88. 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 37028 Closed form for a1dd 47. Proposition 25 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (𝜓 → (𝜃𝜒))))
 
Theoremfrege18 37029 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 37030 A closed form of com45 94. 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 37031 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 37032 A closed form of com3l 86. Proposition 17 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜓 → (𝜒 → (𝜑𝜃))))
 
Theoremfrege13 37033 A closed form of com3r 84. Proposition 13 of [Frege1879] p. 37. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜒 → (𝜑 → (𝜓𝜃))))
 
Theoremfrege14 37034 Closed form of a deduction based on com3r 84. 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 37035 A closed form of syl6 34. Proposition 19 of [Frege1879] p. 39. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜒𝜃) → (𝜑 → (𝜓𝜃))))
 
Theoremfrege23 37036 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 37037 A closed form of com4r 91. 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 37038 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 37039 A closed form of syl8 73. Proposition 20 of [Frege1879] p. 40. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓 → (𝜒𝜃))) → ((𝜃𝜏) → (𝜑 → (𝜓 → (𝜒𝜏)))))
 
20.26.3.4  _Begriffsschrift_ Chapter II Implication and Negation
 
Theoremaxfrege28 37040 Contraposition. Identical to con3 147. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
Axiomax-frege28 37041 Contraposition. Identical to con3 147. 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 37042 Closed form of con3d 146. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
 
Theoremfrege30 37043 Commuted, closed form of con3d 146. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))
 
Theoremaxfrege31 37044 Identical to notnotr 123. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
(¬ ¬ 𝜑𝜑)
 
Axiomax-frege31 37045 𝜑 cannot be denied and (at the same time ) ¬ ¬ 𝜑 affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnotr 123. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)
 
Theoremfrege32 37046 Deduce con1 141 from con3 147. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
 
Theoremfrege33 37047 If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 141. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → (¬ 𝜓𝜑))
 
Theoremfrege34 37048 If as a conseqence of the occurence 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 occurence of the obstacle 𝜓 can be inferred. Closed form of con1d 137. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓)))
 
Theoremfrege35 37049 Commuted, closed form of con1d 137. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (¬ 𝜒 → (𝜑𝜓)))
 
Theoremfrege36 37050 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 pm2.24 119. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (¬ 𝜑𝜓))
 
Theoremfrege37 37051 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 407. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → 𝜒) → (𝜑𝜒))
 
Theoremfrege38 37052 Identical to pm2.21 118. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝜑 → (𝜑𝜓))
 
Theoremfrege39 37053 Syllogism between pm2.18 120 and pm2.24 119. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜑) → (¬ 𝜑𝜓))
 
Theoremfrege40 37054 Anything implies pm2.18 120. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝜑 → ((¬ 𝜓𝜓) → 𝜓))
 
Theoremaxfrege41 37055 Identical to notnot 134. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)
(𝜑 → ¬ ¬ 𝜑)
 
Axiomax-frege41 37056 The affirmation of 𝜑 denies the denial of 𝜑. Identical to notnot 134. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑 → ¬ ¬ 𝜑)
 
Theoremfrege42 37057 Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
¬ ¬ (𝜑𝜑)
 
Theoremfrege43 37058 If there is a choice only between 𝜑 and 𝜑, then 𝜑 takes place. Identical to pm2.18 120. Proposition 43 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜑) → 𝜑)
 
Theoremfrege44 37059 Similar to a commuted pm2.62 423. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜓𝜑) → 𝜑))
 
Theoremfrege45 37060 Deduce pm2.6 180 from con1 141. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → (¬ 𝜓𝜑)) → ((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓)))
 
Theoremfrege46 37061 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 occurences of 𝜑 has 𝜓 as a necessary consequence, then 𝜓 takes place. Identical to pm2.6 180. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theoremfrege47 37062 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 37063 Closed form of syllogism with internal disjunction. If 𝜑 is a sufficient condition for the occurence 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 37175. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → ((𝜒𝜃) → ((𝜓𝜃) → (𝜑𝜃))))
 
Theoremfrege49 37064 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 37065 Closed form of jaoi 392. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜓) → ((¬ 𝜑𝜒) → 𝜓)))
 
Theoremfrege51 37066 Compare with jaod 393. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) → (𝜑 → ((¬ 𝜓𝜃) → 𝜒))))
 
20.26.3.5  _Begriffsschrift_ Chapter II with logical equivalence

Here we leverage df-ifp 1006 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 37067 Justification for ax-frege52a 37068. (Contributed by RP, 17-Apr-2020.)
((𝜑𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
 
Axiomax-frege52a 37068 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 funtion ) and the same proposition, this time where we subsituted 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 37069 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 203. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremfrege53aid 37070 Specialization of frege53a 37071. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → ((𝜑𝜓) → 𝜓))
 
Theoremfrege53a 37071 Lemma for frege55a 37079. 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 37072 Justification for ax-frege54a 37073. Identical to biid 249. (Contributed by RP, 24-Dec-2019.)
(𝜑𝜑)
 
Axiomax-frege54a 37073 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 249. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑𝜑)
 
Theoremfrege54cor0a 37074 Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜓𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
 
Theoremfrege54cor1a 37075 Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if-(𝜑, 𝜑, ¬ 𝜑)
 
Theoremfrege55aid 37076 Lemma for frege57aid 37083. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremfrege55lem1a 37077 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 37078 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 37079 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))
 
Theoremfrege55cor1a 37080 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremfrege56aid 37081 Lemma for frege57aid 37083. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (𝜑𝜓)) → ((𝜓𝜑) → (𝜑𝜓)))
 
Theoremfrege56a 37082 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 37083 This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 216. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremfrege57a 37084 Analogue of frege57aid 37083. 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 37085 Identical to anifp 1013. Justification for ax-frege58a 37086. (Contributed by RP, 28-Mar-2020.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
 
Axiomax-frege58a 37086 If 𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2245. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
 
Theoremfrege58acor 37087 Lemma for frege59a 37088. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓𝜒) ∧ (𝜃𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
 
Theoremfrege59a 37088 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 37024 incorrectly referenced where frege30 37043 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

(if-(𝜑, 𝜓, 𝜃) → (¬ if-(𝜑, 𝜒, 𝜏) → ¬ ((𝜓𝜒) ∧ (𝜃𝜏))))
 
Theoremfrege60a 37089 Swap antecedents of ax-frege58a 37086. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
 
Theoremfrege61a 37090 Lemma for frege65a 37094. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
((if-(𝜑, 𝜓, 𝜒) → 𝜃) → ((𝜓𝜒) → 𝜃))
 
Theoremfrege62a 37091 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2455 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) → (((𝜓𝜒) ∧ (𝜃𝜏)) → if-(𝜑, 𝜒, 𝜏)))
 
Theoremfrege63a 37092 Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(if-(𝜑, 𝜓, 𝜃) → (𝜂 → (((𝜓𝜒) ∧ (𝜃𝜏)) → if-(𝜑, 𝜒, 𝜏))))
 
Theoremfrege64a 37093 Lemma for frege65a 37094. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁))))
 
Theoremfrege65a 37094 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2455 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓𝜒) ∧ (𝜏𝜂)) → (((𝜒𝜃) ∧ (𝜂𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
 
Theoremfrege66a 37095 Swap antecedents of frege65a 37094. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜒𝜃) ∧ (𝜂𝜁)) → (((𝜓𝜒) ∧ (𝜏𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
 
Theoremfrege67a 37096 Lemma for frege68a 37097. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
((((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒))) → (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))))
 
Theoremfrege68a 37097 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.)
(((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))
 
20.26.3.6  _Begriffsschrift_ Chapter II with equivalence of sets
 
Theoremaxfrege52c 37098 Justification for ax-frege52c 37099. (Contributed by RP, 24-Dec-2019.)
(𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
 
Axiomax-frege52c 37099 One side of dfsbcq 3308. 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))
 
Theoremfrege52b 37100 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 subsitute 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))
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40200 403 40201-40300 404 40301-40400 405 40401-40500 406 40501-40600 407 40601-40700 408 40701-40800 409 40801-40900 410 40901-41000 411 41001-41100 412 41101-41200 413 41201-41300 414 41301-41400 415 41401-41500 416 41501-41600 417 41601-41700 418 41701-41800 419 41801-41900 420 41901-42000 421 42001-42100 422 42101-42200 423 42201-42300 424 42301-42400 425 42401-42426
  Copyright terms: Public domain < Previous  Next >