Theorem frege65b 38204

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2563 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

Step	Hyp	Ref	Expression
1		sbim 2395	. . 3 |- ( [ y / x ] ( ph -> ps ) <-> ( [ y / x ] ph -> [ y / x ] ps ) )
2		frege64b 38203	. . 3 |- ( ( [ y / x ] ph -> [ y / x ] ps ) -> ( A. x ( ps -> ch ) -> ( [ y / x ] ph -> [ y / x ] ch ) ) )
3	1, 2	sylbi 207	. 2 |- ( [ y / x ] ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [ y / x ] ph -> [ y / x ] ch ) ) )
4		frege61b 38200	. 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 ) ) ) )
5	3, 4	ax-mp 5	1 |- ( A. x ( ph -> ps ) -> ( A. x ( ps -> ch ) -> ( [ y / x ] ph -> [ y / x ] ch ) ) )

Syntax hints:   -> wi 4   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege58b 38195

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  frege66b  38205