Theorem frege63a 41166

Description: Proposition 63 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege63a	⊢ (if-(𝜑, 𝜓, 𝜃) → (𝜂 → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏))))

Proof of Theorem frege63a

Step	Hyp	Ref	Expression
1		frege62a 41165	. 2 ⊢ (if-(𝜑, 𝜓, 𝜃) → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏)))
2		frege24 41100	. 2 ⊢ ((if-(𝜑, 𝜓, 𝜃) → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏))) → (if-(𝜑, 𝜓, 𝜃) → (𝜂 → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏)))))
3	1, 2	ax-mp 5	1 ⊢ (if-(𝜑, 𝜓, 𝜃) → (𝜂 → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏))))

Syntax hints:   → wi 4   ∧ wa 399  if-wif 1063

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 41075  ax-frege2 41076  ax-frege8 41094  ax-frege58a 41160

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-ifp 1064

This theorem is referenced by: (None)