Theorem frege53a 44441

Description: Lemma for frege55a 44449. Proposition 53 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege53a	⊢ (if-(𝜑, 𝜃, 𝜒) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜃, 𝜒)))

Proof of Theorem frege53a

Step	Hyp	Ref	Expression
1		ax-frege52a 44438	. 2 ⊢ ((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
2		ax-frege8 44390	. 2 ⊢ (((𝜑 ↔ 𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒))) → (if-(𝜑, 𝜃, 𝜒) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜃, 𝜒))))
3	1, 2	ax-mp 5	1 ⊢ (if-(𝜑, 𝜃, 𝜒) → ((𝜑 ↔ 𝜓) → if-(𝜓, 𝜃, 𝜒)))

Syntax hints:   → wi 4   ↔ wb 208  if-wif 1074

This theorem was proved from axioms:  ax-mp 5  ax-frege8 44390  ax-frege52a 44438

This theorem is referenced by:  frege55a  44449