Theorem frege56aid 44624

Description: Lemma for frege57aid 44626. Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege56aid	⊢ (((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) → ((𝜓 ↔ 𝜑) → (𝜑 → 𝜓)))

Proof of Theorem frege56aid

Step	Hyp	Ref	Expression
1		frege55aid 44619	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓))
2		frege9 44566	. 2 ⊢ (((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓)) → (((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) → ((𝜓 ↔ 𝜑) → (𝜑 → 𝜓))))
3	1, 2	ax-mp 5	1 ⊢ (((𝜑 ↔ 𝜓) → (𝜑 → 𝜓)) → ((𝜓 ↔ 𝜑) → (𝜑 → 𝜓)))

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 44544  ax-frege2 44545  ax-frege8 44563

This proof depends on definitions:  df-bi 210

This theorem is used by:  frege57aid  44626