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Theorem frege65b 36577
Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2412 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.)

Assertion
Ref Expression
frege65b  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ( ps  ->  ch )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ch ) ) )

Proof of Theorem frege65b
StepHypRef Expression
1 sbim 2244 . . 3  |-  ( [ y  /  x ]
( ph  ->  ps )  <->  ( [ y  /  x ] ph  ->  [ y  /  x ] ps )
)
2 frege64b 36576 . . 3  |-  ( ( [ y  /  x ] ph  ->  [ y  /  x ] ps )  ->  ( A. x ( ps  ->  ch )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ch ) ) )
31, 2sylbi 200 . 2  |-  ( [ y  /  x ]
( ph  ->  ps )  ->  ( A. x ( ps  ->  ch )  ->  ( [ y  /  x ] ph  ->  [ y  /  x ] ch ) ) )
4 frege61b 36573 . 2  |-  ( ( [ y  /  x ] ( ph  ->  ps )  ->  ( A. x ( ps  ->  ch )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ch ) ) )  ->  ( A. x ( ph  ->  ps )  ->  ( A. x ( ps  ->  ch )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ch ) ) ) )
53, 4ax-mp 5 1  |-  ( A. x ( ph  ->  ps )  ->  ( A. x ( ps  ->  ch )  ->  ( [
y  /  x ] ph  ->  [ y  /  x ] ch ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1450   [wsb 1805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-12 1950  ax-13 2104  ax-frege1 36457  ax-frege2 36458  ax-frege8 36476  ax-frege58b 36568
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-ex 1672  df-nf 1676  df-sb 1806
This theorem is referenced by:  frege66b  36578
  Copyright terms: Public domain W3C validator