Theorem frege64a 37196

Description: Lemma for frege65a 37197. 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 37194	. 2 ⊢ (if-(𝜎, 𝜒, 𝜂) → (((𝜒 → 𝜃) ∧ (𝜂 → 𝜁)) → if-(𝜎, 𝜃, 𝜁)))
2		frege18 37132	. 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 383  if-wif 1006

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 37104  ax-frege2 37105  ax-frege8 37123  ax-frege58a 37189

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007

This theorem is referenced by:  frege65a  37197