Theorem frege55cor1a 37183

Description: Proposition 55 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege55cor1a	⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))

Proof of Theorem frege55cor1a

Step	Hyp	Ref	Expression
1		frege55a 37182	. 2 ⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))
2		frege55lem1a 37180	. 2 ⊢ (((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑)) → ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195  if-wif 1006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege8 37123  ax-frege28 37144  ax-frege52a 37171  ax-frege54a 37176

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by:  frege56a  37185