Theorem frege55a 40234

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

Assertion

Ref	Expression
frege55a	⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))

Proof of Theorem frege55a

Step	Hyp	Ref	Expression
1		frege54cor1a 40230	. 2 ⊢ if-(𝜑, 𝜑, ¬ 𝜑)
2		frege53a 40226	. 2 ⊢ (if-(𝜑, 𝜑, ¬ 𝜑) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208  if-wif 1057

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege8 40175  ax-frege28 40196  ax-frege52a 40223  ax-frege54a 40228

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ifp 1058

This theorem is referenced by:  frege55cor1a  40235