Theorem frege57aid 43367

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 43353	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜓 → 𝜑))
2		frege56aid 43365	. 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 43285  ax-frege2 43286  ax-frege8 43304  ax-frege52a 43352

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061  df-tru 1536  df-fal 1546

This theorem is referenced by:  frege68a  43381  frege68b  43408  frege68c  43426  frege100  43458