Theorem frege55lem2a 40219

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

Assertion

Ref	Expression
frege55lem2a	⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))

Proof of Theorem frege55lem2a

Step	Hyp	Ref	Expression
1		bicom1 223	. 2 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
2		frege54cor0a 40215	. 2 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
3	1, 2	sylib 220	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-frege28 40182

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

This theorem is referenced by: (None)