Theorem frege56a 38165

Description: Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege56a	|- ( ( ( ph <-> ps ) -> (if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) -> ( ( ps <-> ph ) -> (if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) )

Proof of Theorem frege56a

Step	Hyp	Ref	Expression
1		frege55cor1a 38163	. 2 |- ( ( ps <-> ph ) -> ( ph <-> ps ) )
2		frege9 38106	. 2 |- ( ( ( ps <-> ph ) -> ( ph <-> ps ) ) -> ( ( ( ph <-> ps ) -> (if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) -> ( ( ps <-> ph ) -> (if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) ) )
3	1, 2	ax-mp 5	1 |- ( ( ( ph <-> ps ) -> (if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) -> ( ( ps <-> ph ) -> (if- ( ph , ch , th ) -> if- ( ps , ch , th ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196  if-wif 1012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 38084  ax-frege2 38085  ax-frege8 38103  ax-frege28 38124  ax-frege52a 38151  ax-frege54a 38156

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ifp 1013

This theorem is referenced by:  frege57a  38167