Theorem frege55lem2a 37181

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 210	. . 3 ⊢ ((𝜑 ↔ 𝜓) → (𝜓 ↔ 𝜑))
2		frege54cor0a 37177	. . 3 ⊢ ((𝜓 ↔ 𝜑) ↔ if-(𝜓, 𝜑, ¬ 𝜑))
3	1, 2	sylib 207	. 2 ⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 𝜑))
4	3	idi 2	1 ⊢ ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜑, ¬ 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-frege28 37144

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

This theorem is referenced by: (None)