Theorem frege57aid 42623

Description: This is the all imporant formula which allows to apply Frege-style definitions and explore their consequences. A closed form of biimpri 227. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege57aid	⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))

Proof of Theorem frege57aid

Step	Hyp	Ref	Expression
1		frege52aid 42609	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜓 → 𝜑))
2		frege56aid 42621	. 2 ⊢ (((𝜓 ↔ 𝜑) → (𝜓 → 𝜑)) → ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))

Syntax hints:   → wi 4   ↔ wb 205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 42541  ax-frege2 42542  ax-frege8 42560  ax-frege52a 42608

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ifp 1063  df-tru 1545  df-fal 1555

This theorem is referenced by:  frege68a  42637  frege68b  42664  frege68c  42682  frege100  42714