Theorem frege56aid 41367

Description: Lemma for frege57aid 41369. 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 41362	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓))
2		frege9 41309	. 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)))

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 41287  ax-frege2 41288  ax-frege8 41306

This theorem depends on definitions:  df-bi 206

This theorem is referenced by:  frege57aid  41369