Theorem frege57a 37187

Description: Analogue of frege57aid 37186. 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 37171	. 2 ⊢ ((𝜓 ↔ 𝜑) → (if-(𝜓, 𝜒, 𝜃) → if-(𝜑, 𝜒, 𝜃)))
2		frege56a 37185	. 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 195  if-wif 1006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege28 37144  ax-frege52a 37171  ax-frege54a 37176

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

This theorem is referenced by: (None)