Theorem frege56a 45063

Description: Proposition 56 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege56a	⊢ (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓 ↔ 𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))))

Proof of Theorem frege56a

Step	Hyp	Ref	Expression
1		frege55cor1a 45061	. 2 ⊢ ((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓))
2		frege9 45004	. 2 ⊢ (((𝜓 ↔ 𝜑) → (𝜑 ↔ 𝜓)) → (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓 ↔ 𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃)))))
3	1, 2	ax-mp 5	1 ⊢ (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 𝜒, 𝜃))) → ((𝜓 ↔ 𝜑) → (if-(𝜑, 𝜒, 𝜃) → if-(𝜓, 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 44982  ax-frege2 44983  ax-frege8 45001  ax-frege28 45022  ax-frege52a 45049  ax-frege54a 45054

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

This theorem is used by:  frege57a  45065