Theorem frege57a 44872

Description: Analogue of frege57aid 44871. 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 44856	. 2 ⊢ ((𝜓 ↔ 𝜑) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))
2		frege56a 44870	. 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)))

This proof depends on syntax axioms:   → 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-frege1 44789  ax-frege2 44790  ax-frege8 44808  ax-frege28 44829  ax-frege52a 44856  ax-frege54a 44861

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

This theorem is used by: (None)