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Theorem List for Metamath Proof Explorer - 36301-36400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfrege38 36301 Identical to pm2.21 112. Proposition 38 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( -.  ph  ->  ( ph  ->  ps ) )
 
Theoremfrege39 36302 Syllogism between pm2.18 114 and pm2.24 113. Proposition 39 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( -.  ph  ->  ph )  ->  ( -.  ph 
 ->  ps ) )
 
Theoremfrege40 36303 Anything implies pm2.18 114. Proposition 40 of [Frege1879] p. 46. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( -.  ph  ->  ( ( -.  ps  ->  ps )  ->  ps ) )
 
Theoremaxfrege41 36304 Identical to notnot1 126. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph  ->  -.  -.  ph )
 
Axiomax-frege41 36305 The affirmation of  ph denies the denial of  ph. Identical to notnot1 126. Axiom 41 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
 |-  ( ph  ->  -.  -.  ph )
 
Theoremfrege42 36306 Not not id 23. Proposition 42 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  -.  -.  ( ph  ->  ph )
 
Theoremfrege43 36307 If there is a choice only between  ph and  ph, then  ph takes place. Identical to pm2.18 114. Proposition 43 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( -.  ph  ->  ph )  ->  ph )
 
Theoremfrege44 36308 Similar to a commuted pm2.62 411. Proposition 44 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( -.  ph  ->  ps )  ->  ( ( ps  ->  ph )  ->  ph )
 )
 
Theoremfrege45 36309 Deduce pm2.6 174 from con1 132. Proposition 45 of [Frege1879] p. 47. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ( -.  ph  ->  ps )  ->  ( -.  ps  ->  ph ) ) 
 ->  ( ( -.  ph  ->  ps )  ->  (
 ( ph  ->  ps )  ->  ps ) ) )
 
Theoremfrege46 36310 If  ps holds when  ph occurs as well as when  ph does not occur, then  ps holds. If  ps or  ph occurs and if the occurences of  ph has  ps as a necessary consequence, then  ps takes place. Identical to pm2.6 174. Proposition 46 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( -.  ph  ->  ps )  ->  ( ( ph  ->  ps )  ->  ps )
 )
 
Theoremfrege47 36311 Deduce consequence follows from either path implied by a disjunction. If  ph, as well as  ps is sufficient condition for  ch and  ps or  ph takes place, then the proposition  ch holds. Proposition 47 of [Frege1879] p. 48. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( -.  ph  ->  ps )  ->  ( ( ps  ->  ch )  ->  (
 ( ph  ->  ch )  ->  ch ) ) )
 
Theoremfrege48 36312 Closed form of syllogism with internal disjunction. If  ph is a sufficient condition for the occurence of  ch or 
ps and if  ch, as well as  ps, is a sufficient condition for  th, then  ph is a sufficient condition for 
th. See application in frege101 36424. Proposition 48 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  ->  ( -. 
 ps  ->  ch ) )  ->  ( ( ch  ->  th )  ->  ( ( ps  ->  th )  ->  ( ph  ->  th ) ) ) )
 
Theoremfrege49 36313 Closed form of deduction with disjunction. Proposition 49 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( -.  ph  ->  ps )  ->  ( ( ph  ->  ch )  ->  (
 ( ps  ->  ch )  ->  ch ) ) )
 
Theoremfrege50 36314 Closed form of jaoi 381. Proposition 50 of [Frege1879] p. 49. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  ->  ps )  ->  ( ( ch  ->  ps )  ->  ( ( -.  ph  ->  ch )  ->  ps ) ) )
 
Theoremfrege51 36315 Compare with jaod 382. Proposition 51 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  ->  ( ps 
 ->  ch ) )  ->  ( ( th  ->  ch )  ->  ( ph  ->  ( ( -.  ps  ->  th )  ->  ch )
 ) ) )
 
21.25.3.5  _Begriffsschrift_ Chapter II with logical equivalence

Here we leverage df-ifp 1422 to partition a wff into two that are disjoint with the selector wff.

Thus if we are given  |-  ( ph  <-> if- ( ps ,  ch ,  th )
) then we replace the concept (illegal in our notation )  (
ph `  ps ) with if- ( ps ,  ch ,  th ) to reason about the values of the "function." Likewise, we replace the similarly illegal concept  A. ps ph with  ( ch  /\  th ).

 
Theoremaxfrege52a 36316 Justification for ax-frege52a 36317. (Contributed by RP, 17-Apr-2020.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ph ,  th ,  ch )  -> if- ( ps ,  th ,  ch ) ) )
 
Axiomax-frege52a 36317 The case when the content of 
ph is identical with the content of  ps and in which a proposition controlled by an element for which we substitute the content of 
ph is affirmed ( in this specific case the identity logical funtion ) and the same proposition, this time where we subsituted the content of  ps, is denied does not take place. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ph ,  th ,  ch )  -> if- ( ps ,  th ,  ch ) ) )
 
Theoremfrege52aid 36318 The case when the content of 
ph is identical with the content of  ps and in which  ph is affirmed and  ps is denied does not take place. Identical to biimp 197. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  ( ph  ->  ps ) )
 
Theoremfrege53aid 36319 Specialization of frege53a 36320. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( ph  ->  ( ( ph  <->  ps )  ->  ps ) )
 
Theoremfrege53a 36320 Lemma for frege55a 36328. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (if- ( ph ,  th ,  ch )  ->  ( (
 ph 
 <->  ps )  -> if- ( ps ,  th ,  ch ) ) )
 
Theoremaxfrege54a 36321 Justification for ax-frege54a 36322. Identical to biid 240. (Contributed by RP, 24-Dec-2019.)
 |-  ( ph 
 <-> 
 ph )
 
Axiomax-frege54a 36322 Reflexive equality of wffs. The content of  ph is identical with the content of  ph. Part of Axiom 54 of [Frege1879] p. 50. Identical to biid 240. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
 |-  ( ph 
 <-> 
 ph )
 
Theoremfrege54cor0a 36323 Synonym for logical equivalence. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ps  <->  ph )  <-> if- ( ps ,  ph ,  -.  ph ) )
 
Theoremfrege54cor1a 36324 Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |- if- ( ph ,  ph ,  -.  ph )
 
Theoremfrege55aid 36325 Lemma for frege57aid 36332. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  <->  ps )  ->  ( ps 
 <-> 
 ph ) )
 
Theoremfrege55lem1a 36326 Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ta  -> if- ( ps ,  ph ,  -.  ph ) )  ->  ( ta  ->  ( ps  <->  ph ) ) )
 
Theoremfrege55lem2a 36327 Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  <->  ps )  -> if- ( ps ,  ph ,  -.  ph ) )
 
Theoremfrege55a 36328 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  <->  ps )  -> if- ( ps ,  ph ,  -.  ph ) )
 
Theoremfrege55cor1a 36329 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  ( ps 
 <-> 
 ph ) )
 
Theoremfrege56aid 36330 Lemma for frege57aid 36332. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ( ph  <->  ps )  ->  ( ph  ->  ps ) )  ->  ( ( ps  <->  ph )  ->  ( ph  ->  ps ) ) )
 
Theoremfrege56a 36331 Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ( ph  <->  ps )  ->  (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) ) )  ->  ( ( ps  <->  ph )  ->  (if- ( ph ,  ch ,  th )  -> if- ( ps ,  ch ,  th ) ) ) )
 
Theoremfrege57aid 36332 This is the all imporant formula which allows us to apply Frege-style definitions and explore their consequences. A closed form of biimpri 210. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  ( ps  ->  ph ) )
 
Theoremfrege57a 36333 Analogue of frege57aid 36332. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ph  <->  ps )  ->  (if- ( ps ,  ch ,  th )  -> if- ( ph ,  ch ,  th )
 ) )
 
Theoremaxfrege58a 36334 Identical to anifp 1429. Justification for ax-frege58a 36335. (Contributed by RP, 28-Mar-2020.)
 |-  (
 ( ps  /\  ch )  -> if- ( ph ,  ps ,  ch ) )
 
Axiomax-frege58a 36335 If  A. x ph is affirmed,  [
y  /  x ] ph cannot be denied. Identical to stdpc4 2148. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
 |-  (
 ( ps  /\  ch )  -> if- ( ph ,  ps ,  ch ) )
 
Theoremfrege58acor 36336 Lemma for frege59a 36337. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
 |-  (
 ( ( ps  ->  ch )  /\  ( th  ->  ta ) )  ->  (if- ( ph ,  ps ,  th )  -> if- ( ph ,  ch ,  ta )
 ) )
 
Theoremfrege59a 36337 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 36273 incorrectly referenced where frege30 36292 is in the original. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

 |-  (if- ( ph ,  ps ,  th )  ->  ( -. if-
 ( ph ,  ch ,  ta )  ->  -.  (
 ( ps  ->  ch )  /\  ( th  ->  ta )
 ) ) )
 
Theoremfrege60a 36338 Swap antecedents of ax-frege58a 36335. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
 |-  (
 ( ( ps  ->  ( ch  ->  th )
 )  /\  ( ta  ->  ( et  ->  ze )
 ) )  ->  (if- ( ph ,  ch ,  et )  ->  (if- (
 ph ,  ps ,  ta )  -> if- ( ph ,  th ,  ze )
 ) ) )
 
Theoremfrege61a 36339 Lemma for frege65a 36343. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
 |-  (
 (if- ( ph ,  ps ,  ch )  ->  th )  ->  ( ( ps  /\  ch )  ->  th ) )
 
Theoremfrege62a 36340 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2362 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- ( ph ,  ps ,  th )  ->  ( ( ( ps  ->  ch )  /\  ( th  ->  ta )
 )  -> if- ( ph ,  ch ,  ta ) ) )
 
Theoremfrege63a 36341 Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
 |-  (if- ( ph ,  ps ,  th )  ->  ( et 
 ->  ( ( ( ps 
 ->  ch )  /\  ( th  ->  ta ) )  -> if- ( ph ,  ch ,  ta ) ) ) )
 
Theoremfrege64a 36342 Lemma for frege65a 36343. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
 |-  (
 (if- ( ph ,  ps ,  ta )  -> if- ( si ,  ch ,  et ) )  ->  (
 ( ( ch  ->  th )  /\  ( et 
 ->  ze ) )  ->  (if- ( ph ,  ps ,  ta )  -> if- ( si ,  th ,  ze )
 ) ) )
 
Theoremfrege65a 36343 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2362 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
 |-  (
 ( ( ps  ->  ch )  /\  ( ta 
 ->  et ) )  ->  ( ( ( ch 
 ->  th )  /\  ( et  ->  ze ) )  ->  (if- ( ph ,  ps ,  ta )  -> if- ( ph ,  th ,  ze )
 ) ) )
 
Theoremfrege66a 36344 Swap antecedents of frege65a 36343. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
 |-  (
 ( ( ch  ->  th )  /\  ( et 
 ->  ze ) )  ->  ( ( ( ps 
 ->  ch )  /\  ( ta  ->  et ) ) 
 ->  (if- ( ph ,  ps ,  ta )  -> if- ( ph ,  th ,  ze ) ) ) )
 
Theoremfrege67a 36345 Lemma for frege68a 36346. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
 |-  (
 ( ( ( ps 
 /\  ch )  <->  th )  ->  ( th  ->  ( ps  /\  ch ) ) )  ->  ( ( ( ps 
 /\  ch )  <->  th )  ->  ( th  -> if- ( ph ,  ps ,  ch ) ) ) )
 
Theoremfrege68a 36346 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.)
 |-  (
 ( ( ps  /\  ch )  <->  th )  ->  ( th  -> if- ( ph ,  ps ,  ch ) ) )
 
21.25.3.6  _Begriffsschrift_ Chapter II with equivalence of sets
 
Theoremaxfrege52c 36347 Justification for ax-frege52c 36348. (Contributed by RP, 24-Dec-2019.)
 |-  ( A  =  B  ->  (
 [. A  /  x ].
 ph  ->  [. B  /  x ].
 ph ) )
 
Axiomax-frege52c 36348 One side of dfsbcq 3302. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
 |-  ( A  =  B  ->  (
 [. A  /  x ].
 ph  ->  [. B  /  x ].
 ph ) )
 
Theoremfrege52b 36349 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 ] ph,  x can also occur in other than the argument ( z) places. Hence  x may still be contained in  [ y  / 
z ] ph. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )
 
Theoremfrege53b 36350 Lemma for frege102 (via frege92 36415). Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( [ x  /  y ] ph  ->  ( x  =  z  ->  [ z  /  y ] ph )
 )
 
Theoremaxfrege54c 36351 Reflexive equality of classes. Identical to eqid 2423. Justification for ax-frege54c 36352. (Contributed by RP, 24-Dec-2019.)
 |-  A  =  A
 
Axiomax-frege54c 36352 Reflexive equality of sets (as classes). Part of Axiom 54 of [Frege1879] p. 50. Identical to eqid 2423. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
 |-  A  =  A
 
Theoremfrege54b 36353 Reflexive equality of sets. The content of  x is identical with the content of  x. Part of Axiom 54 of [Frege1879] p. 50. Slightly specialized version of eqid 2423. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  x  =  x
 
Theoremfrege54cor1b 36354 Reflexive equality. (Contributed by RP, 24-Dec-2019.)
 |-  [ x  /  y ] y  =  x
 
Theoremfrege55lem1b 36355* Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  [ x  /  y ] y  =  z )  ->  ( ph  ->  x  =  z ) )
 
Theoremfrege55lem2b 36356 Lemma for frege55b 36357. Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  [ y  /  z ]
 z  =  x )
 
Theoremfrege55b 36357 Lemma for frege57b 36359. Proposition 55 of [Frege1879] p. 50.

Note that eqtr2 2450 incorporates eqcom 2432 which is stronger than this proposition which is identical to equcomi 1844. Is is possible that Frege tricked himself into assuming what he was out to prove? (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

 |-  ( x  =  y  ->  y  =  x )
 
Theoremfrege56b 36358 Lemma for frege57b 36359. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( x  =  y 
 ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph )
 )  ->  ( y  =  x  ->  ( [ x  /  z ] ph  ->  [ y  /  z ] ph ) ) )
 
Theoremfrege57b 36359 Analogue of frege57aid 36332. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( x  =  y  ->  ( [ y  /  z ] ph  ->  [ x  /  z ] ph )
 )
 
Theoremaxfrege58b 36360 If  A. x ph is affirmed,  [
y  /  x ] ph cannot be denied. Identical to stdpc4 2148. Justification for ax-frege58b 36361. (Contributed by RP, 28-Mar-2020.)
 |-  ( A. x ph  ->  [ y  /  x ] ph )
 
Axiomax-frege58b 36361 If  A. x ph is affirmed,  [
y  /  x ] ph cannot be denied. Identical to stdpc4 2148. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (New usage is discouraged.)
 |-  ( A. x ph  ->  [ y  /  x ] ph )
 
Theoremfrege58bid 36362 If  A. x ph is affirmed,  ph cannot be denied. Identical to sp 1911. See ax-frege58b 36361 and frege58c 36381 for versions which more closely track the original. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
 |-  ( A. x ph  ->  ph )
 
Theoremfrege58bcor 36363 Lemma for frege59b 36364. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( A. x ( ph  ->  ps )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ps ) )
 
Theoremfrege59b 36364 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 36273 incorrectly referenced where frege30 36292 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

 |-  ( [ x  /  y ] ph  ->  ( -.  [ x  /  y ] ps  ->  -.  A. y (
 ph  ->  ps ) ) )
 
Theoremfrege60b 36365 Swap antecedents of ax-frege58b 36361. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( A. x ( ph  ->  ( ps  ->  ch )
 )  ->  ( [
 y  /  x ] ps  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ch ) ) )
 
Theoremfrege61b 36366 Lemma for frege65b 36370. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( [ x  /  y ] ph  ->  ps )  ->  ( A. y ph  ->  ps ) )
 
Theoremfrege62b 36367 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2362 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( [ x  /  y ] ph  ->  ( A. y ( ph  ->  ps )  ->  [ x  /  y ] ps )
 )
 
Theoremfrege63b 36368 Lemma for frege91 36414. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( [ x  /  y ] ph  ->  ( ps  ->  ( A. y (
 ph  ->  ch )  ->  [ x  /  y ] ch )
 ) )
 
Theoremfrege64b 36369 Lemma for frege65b 36370. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( [ x  /  y ] ph  ->  [ z  /  y ] ps )  ->  ( A. y ( ps  ->  ch )  ->  ( [ x  /  y ] ph  ->  [ z  /  y ] ch )
 ) )
 
Theoremfrege65b 36370 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2362 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53.

In Frege care is taken to point out that the variables in the first clauses are independent of each other and of the final term so another valid translation could be :  |-  ( A. x ( [ x  /  a ] ph  ->  [ x  /  b ] ps )  ->  ( A. y ( [ y  /  b ] ps  ->  [ y  /  c ] ch )  ->  ( [ z  /  a ] ph  ->  [ z  /  c ] ch ) ) ). But that is perhaps too pedantic a translation for this exploration. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

 |-  ( A. x ( ph  ->  ps )  ->  ( A. x ( ps  ->  ch )  ->  ( [
 y  /  x ] ph  ->  [ y  /  x ] ch ) ) )
 
Theoremfrege66b 36371 Swap antecedents of frege65b 36370. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( A. x ( ph  ->  ps )  ->  ( A. x ( ch  ->  ph )  ->  ( [
 y  /  x ] ch  ->  [ y  /  x ] ps ) ) )
 
Theoremfrege67b 36372 Lemma for frege68b 36373. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( ( A. x ph  <->  ps )  ->  ( ps  ->  A. x ph )
 )  ->  ( ( A. x ph  <->  ps )  ->  ( ps  ->  [ y  /  x ] ph ) ) )
 
Theoremfrege68b 36373 Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  (
 ( A. x ph  <->  ps )  ->  ( ps  ->  [ y  /  x ] ph ) )
 
21.25.3.7  _Begriffsschrift_ Chapter II with equivalence of classes (where they are sets)
 
Theoremfrege53c 36374 Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( [. A  /  x ].
 ph  ->  ( A  =  B  ->  [. B  /  x ].
 ph ) )
 
Theoremfrege54cor1c 36375* Reflexive equality. (Contributed by RP, 24-Dec-2019.) (Revised by RP, 25-Apr-2020.)
 |-  A  e.  C   =>    |-  [. A  /  x ]. x  =  A
 
Theoremfrege55lem1c 36376* Necessary deduction regarding substitution of value in equality. (Contributed by RP, 24-Dec-2019.)
 |-  (
 ( ph  ->  [. A  /  x ]. x  =  B )  ->  ( ph  ->  A  =  B ) )
 
Theoremfrege55lem2c 36377* Core proof of Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( x  =  A  ->  [. A  /  z ]. z  =  x )
 
Theoremfrege55c 36378 Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  ( x  =  A  ->  A  =  x )
 
Theoremfrege56c 36379* Lemma for frege57c 36380. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  B  e.  C   =>    |-  ( ( A  =  B  ->  ( [. A  /  x ]. ph  ->  [. B  /  x ]. ph ) )  ->  ( B  =  A  ->  (
 [. A  /  x ].
 ph  ->  [. B  /  x ].
 ph ) ) )
 
Theoremfrege57c 36380* Swap order of implication in ax-frege52c 36348. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  C   =>    |-  ( A  =  B  ->  ( [. B  /  x ]. ph  ->  [. A  /  x ]. ph )
 )
 
Theoremfrege58c 36381 Principle related to sp 1911. Axiom 58 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( A. x ph  -> 
 [. A  /  x ].
 ph )
 
Theoremfrege59c 36382 A kind of Aristotelian inference. Proposition 59 of [Frege1879] p. 51.

Note: in the Bauer-Meenfelberg translation published in van Heijenoort's collection From Frege to Goedel, this proof has the frege12 36273 incorrectly referenced where frege30 36292 is in the original. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

 |-  A  e.  B   =>    |-  ( [. A  /  x ]. ph  ->  ( -.  [. A  /  x ].
 ps  ->  -.  A. x (
 ph  ->  ps ) ) )
 
Theoremfrege60c 36383 Swap antecedents of frege58c 36381. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( A. x (
 ph  ->  ( ps  ->  ch ) )  ->  ( [. A  /  x ].
 ps  ->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ch ) ) )
 
Theoremfrege61c 36384 Lemma for frege65c 36388. Proposition 61 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( ( [. A  /  x ]. ph  ->  ps )  ->  ( A. x ph  ->  ps )
 )
 
Theoremfrege62c 36385 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2362 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( [. A  /  x ]. ph  ->  ( A. x ( ph  ->  ps )  ->  [. A  /  x ]. ps ) )
 
Theoremfrege63c 36386 Analogue of frege63b 36368. Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( [. A  /  x ]. ph  ->  ( ps 
 ->  ( A. x (
 ph  ->  ch )  ->  [. A  /  x ]. ch )
 ) )
 
Theoremfrege64c 36387 Lemma for frege65c 36388. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( ( [. C  /  x ]. ph  ->  [. A  /  x ]. ps )  ->  ( A. x ( ps  ->  ch )  ->  ( [. C  /  x ]. ph  ->  [. A  /  x ]. ch ) ) )
 
Theoremfrege65c 36388 A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2362 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ( ps  ->  ch )  ->  ( [. A  /  x ]. ph  ->  [. A  /  x ]. ch ) ) )
 
Theoremfrege66c 36389 Swap antecedents of frege65c 36388. Proposition 66 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( A. x (
 ph  ->  ps )  ->  ( A. x ( ch  ->  ph )  ->  ( [. A  /  x ]. ch  -> 
 [. A  /  x ].
 ps ) ) )
 
Theoremfrege67c 36390 Lemma for frege68c 36391. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( ( ( A. x ph  <->  ps )  ->  ( ps  ->  A. x ph )
 )  ->  ( ( A. x ph  <->  ps )  ->  ( ps  ->  [. A  /  x ].
 ph ) ) )
 
Theoremfrege68c 36391 Combination of applying a definition and applying it to a specific instance. Proposition 68 of [Frege1879] p. 54. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)
 |-  A  e.  B   =>    |-  ( ( A. x ph  <->  ps )  ->  ( ps  -> 
 [. A  /  x ].
 ph ) )
 
21.25.3.8  _Begriffsschrift_ Chapter III Properties hereditary in a sequence

 ( R " A )  C_  A means membership in  A is hereditary in the sequence dictated by relation  R. This differs from the set-theoretic notion that a set is hereditary in a property: that all of its elements have a property and all of their elements have the property and so-on.

While the above notation is modern, it is cumbersome in the case when  A is complex and to more closely follow Frege, we abbreviate it with new notation 
R hereditary  A. This greatly shortens the statements for frege97 36420 and frege109 36432.

dffrege69 36392 through frege75 36398 develop this, but translation to Metamath is pending some decisions.

While Frege does not limit discussion to sets, we may have to depart from Frege by limiting  R or  A to sets when we quantify over all hereditary relations or all classes where membership is hereditary in a sequence dictated by  R.

 
Theoremdffrege69 36392* If from the proposition that  x has property  A it can be inferred generally, whatever  x may be, that every result of an application of the procedure  R to  x has property  A, then we say " Property  A is hereditary in the  R-sequence. Definition 69 of [Frege1879] p. 55. (Contributed by RP, 28-Mar-2020.)
 |-  ( A. x ( x  e.  A  ->  A. y ( x R y  ->  y  e.  A )
 ) 
 <->  R hereditary  A )
 
Theoremfrege70 36393* Lemma for frege72 36395. Proposition 70 of [Frege1879] p. 58. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  V   =>    |-  ( R hereditary  A  ->  ( X  e.  A  ->  A. y ( X R y  ->  y  e.  A ) ) )
 
Theoremfrege71 36394* Lemma for frege72 36395. Proposition 71 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 3-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  V   =>    |-  ( ( A. z
 ( X R z 
 ->  z  e.  A )  ->  ( X R Y  ->  Y  e.  A ) )  ->  ( R hereditary  A  ->  ( X  e.  A  ->  ( X R Y  ->  Y  e.  A ) ) ) )
 
Theoremfrege72 36395 If property  A is hereditary in the  R-sequence, if  x has property  A, and if  y is a result of an application of the procedure  R to  x, then  y has property  A. Proposition 72 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   =>    |-  ( R hereditary  A  ->  ( X  e.  A  ->  ( X R Y  ->  Y  e.  A ) ) )
 
Theoremfrege73 36396 Lemma for frege87 36410. Proposition 73 of [Frege1879] p. 59. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   =>    |-  (
 ( R hereditary  A  ->  X  e.  A )  ->  ( R hereditary  A  ->  ( X R Y  ->  Y  e.  A ) ) )
 
Theoremfrege74 36397 If  X has a property  A that is hereditary in the  R-sequence, then every result of a application of the procedure  R to  X has the property  A. Proposition 74 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Revised by RP, 5-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   =>    |-  ( X  e.  A  ->  ( R hereditary  A  ->  ( X R Y  ->  Y  e.  A ) ) )
 
Theoremfrege75 36398* If from the proposition that  x has property  A, whatever  x may be, it can be inferred that every result of an application of the procedure  R to  x has property  A, then property  A is hereditary in the  R-sequence. Proposition 75 of [Frege1879] p. 60. (Contributed by RP, 28-Mar-2020.) (Proof modification is discouraged.)
 |-  ( A. x ( x  e.  A  ->  A. y ( x R y  ->  y  e.  A )
 )  ->  R hereditary  A )
 
21.25.3.9  _Begriffsschrift_ Chapter III Following in a sequence

 p (
t+ `  R
) c means  c follows  p in the  R-sequence.

dffrege76 36399 through frege98 36421 develop this.

This will be shown to be the transitive closure of the relation  R. But more work needs to be done on transitive closure of relations before this is ready for Metamath.

 
Theoremdffrege76 36399* If from the two propositions that every result of an application of the procedure  R to  B has property  f and that property  f is hereditary in the  R-sequence, it can be inferred, whatever  f may be, that  E has property  f, then we say  E follows  B in the  R-sequence. Definition 76 of [Frege1879] p. 60.

Each of  B,  E and  R must be sets. (Contributed by RP, 2-Jul-2020.)

 |-  B  e.  U   &    |-  E  e.  V   &    |-  R  e.  W   =>    |-  ( A. f ( R hereditary  f  ->  ( A. a ( B R a  ->  a  e.  f
 )  ->  E  e.  f ) )  <->  B ( t+ `
  R ) E )
 
Theoremfrege77 36400* If  Y follows  X in the  R-sequence, if property  A is hereditary in the  R-sequence, and if every result of an application of the procedure  R to  X has the property  A, then  Y has property  A. Proposition 77 of [Frege1879] p. 62. (Contributed by RP, 29-Jun-2020.) (Revised by RP, 2-Jul-2020.) (Proof modification is discouraged.)
 |-  X  e.  U   &    |-  Y  e.  V   &    |-  R  e.  W   &    |-  A  e.  B   =>    |-  ( X ( t+ `
  R ) Y 
 ->  ( R hereditary  A  ->  (
 A. a ( X R a  ->  a  e.  A )  ->  Y  e.  A ) ) )
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