Theorem frege60b 36572

Description: Swap antecedents of ax-frege58b 36568. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege60b	|- ( A. x ( ph -> ( ps -> ch ) ) -> ( [ y / x ] ps -> ( [ y / x ] ph -> [ y / x ] ch ) ) )

Proof of Theorem frege60b

Step	Hyp	Ref	Expression
1		ax-frege58b 36568	. . 3 |- ( A. x ( ph -> ( ps -> ch ) ) -> [ y / x ] ( ph -> ( ps -> ch ) ) )
2		sbim 2244	. . . 4 |- ( [ y / x ] ( ph -> ( ps -> ch ) ) <-> ( [ y / x ] ph -> [ y / x ] ( ps -> ch ) ) )
3		sbim 2244	. . . . 5 |- ( [ y / x ] ( ps -> ch ) <-> ( [ y / x ] ps -> [ y / x ] ch ) )
4	3	imbi2i 319	. . . 4 |- ( ( [ y / x ] ph -> [ y / x ] ( ps -> ch ) ) <-> ( [ y / x ] ph -> ( [ y / x ] ps -> [ y / x ] ch ) ) )
5	2, 4	bitri 257	. . 3 |- ( [ y / x ] ( ph -> ( ps -> ch ) ) <-> ( [ y / x ] ph -> ( [ y / x ] ps -> [ y / x ] ch ) ) )
6	1, 5	sylib 201	. 2 |- ( A. x ( ph -> ( ps -> ch ) ) -> ( [ y / x ] ph -> ( [ y / x ] ps -> [ y / x ] ch ) ) )
7		frege12 36480	. 2 |- ( ( A. x ( ph -> ( ps -> ch ) ) -> ( [ y / x ] ph -> ( [ y / x ] ps -> [ y / x ] ch ) ) ) -> ( A. x ( ph -> ( ps -> ch ) ) -> ( [ y / x ] ps -> ( [ y / x ] ph -> [ y / x ] ch ) ) ) )
8	6, 7	ax-mp 5	1 |- ( A. x ( ph -> ( ps -> ch ) ) -> ( [ y / x ] ps -> ( [ y / x ] ph -> [ y / x ] ch ) ) )

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: (None)