Theorem frege64a 41379

Description: Lemma for frege65a 41380. Proposition 64 of [Frege1879] p. 53. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege64a	⊢ ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁))))

Proof of Theorem frege64a

Step	Hyp	Ref	Expression
1		frege62a 41377	. 2 ⊢ (if-(𝜎, 𝜒, 𝜂) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → if-(𝜎, 𝜃, 𝜁)))
2		frege18 41315	. 2 ⊢ ((if-(𝜎, 𝜒, 𝜂) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → if-(𝜎, 𝜃, 𝜁))) → ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁)))))
3	1, 2	ax-mp 5	1 ⊢ ((if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜒, 𝜂)) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜎, 𝜃, 𝜁))))

Syntax hints:   → wi 4   ∧ wa 395  if-wif 1059

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 41287  ax-frege2 41288  ax-frege8 41306  ax-frege58a 41372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-ifp 1060

This theorem is referenced by:  frege65a  41380