Theorem frege65a 43344

Description: A kind of Aristotelian inference. This judgement replaces the mode of inference barbara 2654 when the minor premise has a general context. Proposition 65 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege65a	⊢ (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))

Proof of Theorem frege65a

Step	Hyp	Ref	Expression
1		ifpimim 42970	. . 3 ⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜏 → 𝜂)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜒, 𝜂)))
2		frege64a 43343	. . 3 ⊢ ((if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
3	1, 2	syl 17	. 2 ⊢ (if-(𝜑, (𝜓 → 𝜒), (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
4		frege61a 43340	. 2 ⊢ ((if-(𝜑, (𝜓 → 𝜒), (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))) → (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))))
5	3, 4	ax-mp 5	1 ⊢ (((𝜓 → 𝜒) ∧ (𝜏 → 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))

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-frege1 43251  ax-frege2 43252  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:  frege66a  43345