Theorem frege60c 38217

Description: Swap antecedents of frege58c 38215. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
frege59c.a	|- A e. B

Assertion

Ref	Expression
frege60c	|- ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ps -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )

Proof of Theorem frege60c

Step	Hyp	Ref	Expression
1		frege59c.a	. . . 4 |- A e. B
2	1	frege58c 38215	. . 3 |- ( A. x ( ph -> ( ps -> ch ) ) -> [. A / x ]. ( ph -> ( ps -> ch ) ) )
3		sbcim1 3482	. . . 4 |- ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> [. A / x ]. ( ps -> ch ) ) )
4		sbcim1 3482	. . . 4 |- ( [. A / x ]. ( ps -> ch ) -> ( [. A / x ]. ps -> [. A / x ]. ch ) )
5	3, 4	syl6 35	. . 3 |- ( [. A / x ]. ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
6	2, 5	syl 17	. 2 |- ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) )
7		frege12 38107	. 2 |- ( ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ph -> ( [. A / x ]. ps -> [. A / x ]. ch ) ) ) -> ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ps -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) ) )
8	6, 7	ax-mp 5	1 |- ( A. x ( ph -> ( ps -> ch ) ) -> ( [. A / x ]. ps -> ( [. A / x ]. ph -> [. A / x ]. ch ) ) )

Syntax hints:   -> wi 4   A.wal 1481   e. wcel 1990   [.wsbc 3435

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-9 1999  ax-10 2019  ax-12 2047  ax-13 2246  ax-ext 2602  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-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-v 3202  df-sbc 3436

This theorem is referenced by:  frege93  38250