Theorem frege57aid 45064

Description: This is the all important formula which allows to apply Frege-style definitions and explore their consequences. A closed form of biimpri 231. 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 45050	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜓 → 𝜑))
2		frege56aid 45062	. 2 ⊢ (((𝜓 ↔ 𝜑) → (𝜓 → 𝜑)) → ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 → 𝜑))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 44982  ax-frege2 44983  ax-frege8 45001  ax-frege52a 45049

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ifp 1079  df-tru 1573  df-fal 1583

This theorem is used by:  frege68a  45078  frege68b  45105  frege68c  45123  frege100  45155