Theorem frege60a 39012

Description: Swap antecedents of ax-frege58a 39009. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)

Assertion

Ref	Expression
frege60a	⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))

Proof of Theorem frege60a

Step	Hyp	Ref	Expression
1		frege58acor 39010	. . 3 ⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, (𝜒 → 𝜃), (𝜂 → 𝜁))))
2		ifpimim 38696	. . 3 ⊢ (if-(𝜑, (𝜒 → 𝜃), (𝜂 → 𝜁)) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁)))
3	1, 2	syl6 35	. 2 ⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜓, 𝜏) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁))))
4		frege12 38947	. 2 ⊢ ((((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜓, 𝜏) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁)))) → (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))))
5	3, 4	ax-mp 5	1 ⊢ (((𝜓 → (𝜒 → 𝜃)) ∧ (𝜏 → (𝜂 → 𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))

Syntax hints:   → wi 4   ∧ wa 386  if-wif 1091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 38924  ax-frege2 38925  ax-frege8 38943  ax-frege58a 39009

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ifp 1092

This theorem is referenced by: (None)