Theorem frege55a 45058

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 45054	. 2 ⊢ if-(𝜑, 𝜑, ¬ 𝜑)
2		frege53a 45050	. 2 ⊢ (if-(𝜑, 𝜑, ¬ 𝜑) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑)))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 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 44999  ax-frege28 45020  ax-frege52a 45047  ax-frege54a 45052

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

This theorem is used by:  frege55cor1a  45059