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Theorem List for Metamath Proof Explorer - 38401-38500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
20.27.3.3  _Begriffsschrift_ Chapter II Implication
 
Axiomax-frege1 38401 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 38402 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 38403 Simplification of triple conjunction. Compare with simp2 1082. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓 → (𝜒𝜓)))
 
Theoremrp-simp2 38404 Simplification of triple conjunction. Identical to simp2 1082. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓𝜒) → 𝜓)
 
Theoremrp-frege3g 38405 Add antecedent to ax-frege2 38402. More general statement than frege3 38406. Like ax-frege2 38402, 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 38402 in Metamath. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

(𝜑 → ((𝜓 → (𝜒𝜃)) → ((𝜓𝜒) → (𝜓𝜃))))
 
Theoremfrege3 38406 Add antecedent to ax-frege2 38402. Special case of rp-frege3g 38405. Proposition 3 of [Frege1879] p. 29. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒 → (𝜑𝜓)) → ((𝜒𝜑) → (𝜒𝜓))))
 
Theoremrp-misc1-frege 38407 Double-use of ax-frege2 38402. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑 → (𝜓𝜒)) → (𝜑𝜓)) → ((𝜑 → (𝜓𝜒)) → (𝜑𝜒)))
 
Theoremrp-frege24 38408 Introducing an embedded antecedent. Alternate proof for frege24 38426. Closed form for a1d 25. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))
 
Theoremrp-frege4g 38409 Deduction related to distribution. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → (𝜒𝜃))) → (𝜑 → ((𝜓𝜒) → (𝜓𝜃))))
 
Theoremfrege4 38410 Special case of closed form of a2d 29. Special case of rp-frege4g 38409. Proposition 4 of [Frege1879] p. 31. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (𝜒 → (𝜑𝜓))) → ((𝜑𝜓) → ((𝜒𝜑) → (𝜒𝜓))))
 
Theoremfrege5 38411 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 38412 Distribute antecedent and add another. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓𝜒)) → (𝜃 → ((𝜑𝜓) → (𝜑𝜒))))
 
Theoremrp-4frege 38413 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
((𝜑 → ((𝜓𝜑) → 𝜒)) → (𝜑𝜒))
 
Theoremrp-6frege 38414 Elimination of a nested antecedent of special form. (Contributed by RP, 24-Dec-2019.)
(𝜑 → ((𝜓 → ((𝜒𝜓) → 𝜃)) → (𝜓𝜃)))
 
Theoremrp-8frege 38415 Eliminate antecedent when it is implied by previous antecedent. (Contributed by RP, 24-Dec-2019.)
((𝜑 → (𝜓 → ((𝜒𝜓) → 𝜃))) → (𝜑 → (𝜓𝜃)))
 
Theoremrp-frege25 38416 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 38417 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 38418 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 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 38419 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 38420 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 38401, and ax-frege2 38402. (Redundant) Axiom 8 of [Frege1879] p. 35. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (𝜑𝜒)))
 
Theoremfrege26 38421 Identical to idd 24. Proposition 26 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (𝜓𝜓))
 
Theoremfrege27 38422 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 38423 Closed form of syl 17 with swapped antecedents. This proposition differs from frege5 38411 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 38424 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 38425 Elimination of a nested antecedent as a partial converse of ja 173. 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 38426 Closed form for a1d 25. Deduction introducing an embedded antecedent. Identical to rp-frege24 38408 which was proved without relying on ax-frege8 38420. Proposition 24 of [Frege1879] p. 42. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜑 → (𝜒𝜓)))
 
Theoremfrege16 38427 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 38428 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 38429 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 38430 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 38431 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 38432 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 38433 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 38434 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 38435 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 38436 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 38437 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 38438 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 38439 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)))))
 
20.27.3.4  _Begriffsschrift_ Chapter II Implication and Negation
 
Theoremaxfrege28 38440 Contraposition. Identical to con3 149. Theorem *2.16 of [WhiteheadRussell] p. 103. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (¬ 𝜓 → ¬ 𝜑))
 
Axiomax-frege28 38441 Contraposition. Identical to con3 149. 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 38442 Closed form of con3d 148. Proposition 29 of [Frege1879] p. 43. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜑 → (¬ 𝜒 → ¬ 𝜓)))
 
Theoremfrege30 38443 Commuted, closed form of con3d 148. Proposition 30 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → (𝜓 → (¬ 𝜒 → ¬ 𝜑)))
 
Theoremaxfrege31 38444 Identical to notnotr 125. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.)
(¬ ¬ 𝜑𝜑)
 
Axiomax-frege31 38445 𝜑 cannot be denied and (at the same time ) ¬ ¬ 𝜑 affirmed. Duplex negatio affirmat. The denial of the denial is affirmation. Identical to notnotr 125. Axiom 31 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(¬ ¬ 𝜑𝜑)
 
Theoremfrege32 38446 Deduce con1 143 from con3 149. Proposition 32 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → (¬ 𝜓 → ¬ ¬ 𝜑)) → ((¬ 𝜑𝜓) → (¬ 𝜓𝜑)))
 
Theoremfrege33 38447 If 𝜑 or 𝜓 takes place, then 𝜓 or 𝜑 takes place. Identical to con1 143. Proposition 33 of [Frege1879] p. 44. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → (¬ 𝜓𝜑))
 
Theoremfrege34 38448 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 139. Proposition 34 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (𝜑 → (¬ 𝜒𝜓)))
 
Theoremfrege35 38449 Commuted, closed form of con1d 139. Proposition 35 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → (¬ 𝜒 → (𝜑𝜓)))
 
Theoremfrege36 38450 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 pm2.24 121. Proposition 36 of [Frege1879] p. 45. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → (¬ 𝜑𝜓))
 
Theoremfrege37 38451 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 408. Proposition 37 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → 𝜒) → (𝜑𝜒))
 
Theoremfrege38 38452 Identical to pm2.21 120. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝜑 → (𝜑𝜓))
 
Theoremfrege39 38453 Syllogism between pm2.18 122 and pm2.24 121. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜑) → (¬ 𝜑𝜓))
 
Theoremfrege40 38454 Anything implies pm2.18 122. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
𝜑 → ((¬ 𝜓𝜓) → 𝜓))
 
Theoremaxfrege41 38455 Identical to notnot 136. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)
(𝜑 → ¬ ¬ 𝜑)
 
Axiomax-frege41 38456 The affirmation of 𝜑 denies the denial of 𝜑. Identical to notnot 136. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑 → ¬ ¬ 𝜑)
 
Theoremfrege42 38457 Not not id 22. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
¬ ¬ (𝜑𝜑)
 
Theoremfrege43 38458 If there is a choice only between 𝜑 and 𝜑, then 𝜑 takes place. Identical to pm2.18 122. Proposition 43 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜑) → 𝜑)
 
Theoremfrege44 38459 Similar to a commuted pm2.62 424. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜓𝜑) → 𝜑))
 
Theoremfrege45 38460 Deduce pm2.6 182 from con1 143. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((¬ 𝜑𝜓) → (¬ 𝜓𝜑)) → ((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓)))
 
Theoremfrege46 38461 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 pm2.6 182. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((¬ 𝜑𝜓) → ((𝜑𝜓) → 𝜓))
 
Theoremfrege47 38462 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 38463 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 38575. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (¬ 𝜓𝜒)) → ((𝜒𝜃) → ((𝜓𝜃) → (𝜑𝜃))))
 
Theoremfrege49 38464 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 38465 Closed form of jaoi 393. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → ((𝜒𝜓) → ((¬ 𝜑𝜒) → 𝜓)))
 
Theoremfrege51 38466 Compare with jaod 394. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑 → (𝜓𝜒)) → ((𝜃𝜒) → (𝜑 → ((¬ 𝜓𝜃) → 𝜒))))
 
20.27.3.5  _Begriffsschrift_ Chapter II with logical equivalence

Here we leverage df-ifp 1033 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 38467 Justification for ax-frege52a 38468. (Contributed by RP, 17-Apr-2020.)
((𝜑𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
 
Axiomax-frege52a 38468 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 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 38469 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 205. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜑𝜓))
 
Theoremfrege53aid 38470 Specialization of frege53a 38471. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(𝜑 → ((𝜑𝜓) → 𝜓))
 
Theoremfrege53a 38471 Lemma for frege55a 38479. 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 38472 Justification for ax-frege54a 38473. Identical to biid 251. (Contributed by RP, 24-Dec-2019.)
(𝜑𝜑)
 
Axiomax-frege54a 38473 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 251. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
(𝜑𝜑)
 
Theoremfrege54cor0a 38474 Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜓𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
 
Theoremfrege54cor1a 38475 Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
if-(𝜑, 𝜑, ¬ 𝜑)
 
Theoremfrege55aid 38476 Lemma for frege57aid 38483. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremfrege55lem1a 38477 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 38478 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 38479 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))
 
Theoremfrege55cor1a 38480 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremfrege56aid 38481 Lemma for frege57aid 38483. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
(((𝜑𝜓) → (𝜑𝜓)) → ((𝜓𝜑) → (𝜑𝜓)))
 
Theoremfrege56a 38482 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 38483 This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 218. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
((𝜑𝜓) → (𝜓𝜑))
 
Theoremfrege57a 38484 Analogue of frege57aid 38483. 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 38485 Identical to anifp 1040. Justification for ax-frege58a 38486. (Contributed by RP, 28-Mar-2020.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
 
Axiomax-frege58a 38486 If 𝑥𝜑 is affirmed, [𝑦 / 𝑥]𝜑 cannot be denied. Identical to stdpc4 2381. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
 
Theoremfrege58acor 38487 Lemma for frege59a 38488. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
(((𝜓𝜒) ∧ (𝜃𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
 
Theoremfrege59a 38488 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 38424 incorrectly referenced where frege30 38443 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 38489 Swap antecedents of ax-frege58a 38486. 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 38490 Lemma for frege65a 38494. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
((if-(𝜑, 𝜓, 𝜒) → 𝜃) → ((𝜓𝜒) → 𝜃))
 
Theoremfrege62a 38491 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2592 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 38492 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 38493 Lemma for frege65a 38494. 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 38494 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2592 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 38495 Swap antecedents of frege65a 38494. 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 38496 Lemma for frege68a 38497. 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 38497 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)))
 
20.27.3.6  _Begriffsschrift_ Chapter II with equivalence of sets
 
Theoremaxfrege52c 38498 Justification for ax-frege52c 38499. (Contributed by RP, 24-Dec-2019.)
(𝐴 = 𝐵 → ([𝐴 / 𝑥]𝜑[𝐵 / 𝑥]𝜑))
 
Axiomax-frege52c 38499 One side of dfsbcq 3470. 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 38500 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))
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