Theorem frege57a 43334

Description: Analogue of frege57aid 43333. Proposition 57 of [Frege1879] p. 51. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege57a	⊢ ((𝜑 ↔ 𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))

Proof of Theorem frege57a

Step	Hyp	Ref	Expression
1		ax-frege52a 43318	. 2 ⊢ ((𝜓 ↔ 𝜑) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))
2		frege56a 43332	. 2 ⊢ (((𝜓 ↔ 𝜑) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃))) → ((𝜑 ↔ 𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃))))
3	1, 2	ax-mp 5	1 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))

Syntax hints:   → wi 4   ↔ wb 205  if-wif 1060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 43251  ax-frege2 43252  ax-frege8 43270  ax-frege28 43291  ax-frege52a 43318  ax-frege54a 43323

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061

This theorem is referenced by: (None)