Theorem frege55cor1a 44711

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 44710	. 2 ⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))
2		frege55lem1a 44708	. 2 ⊢ (((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑)) → ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  if-wif 1078

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege8 44651  ax-frege28 44672  ax-frege52a 44699  ax-frege54a 44704

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

This theorem is used by:  frege56a  44713