Theorem frege62a 43341

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2654 when the minor premise has a particular context. Proposition 62 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege62a	⊢ (if-(𝜑, 𝜓, 𝜃) → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏)))

Proof of Theorem frege62a

Step	Hyp	Ref	Expression
1		frege58acor 43337	. 2 ⊢ (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏)))
2		ax-frege8 43270	. 2 ⊢ ((((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → (if-(𝜑, 𝜓, 𝜃) → if-(𝜑, 𝜒, 𝜏))) → (if-(𝜑, 𝜓, 𝜃) → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏))))
3	1, 2	ax-mp 5	1 ⊢ (if-(𝜑, 𝜓, 𝜃) → (((𝜓 → 𝜒) ∧ (𝜃 → 𝜏)) → if-(𝜑, 𝜒, 𝜏)))

Syntax hints:   → wi 4   ∧ wa 394  if-wif 1060

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege8 43270  ax-frege58a 43336

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-ifp 1061

This theorem is referenced by:  frege63a  43342  frege64a  43343