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Theorem frege60a 36995
Description: Swap antecedents of ax-frege58a 36992. Proposition 60 of [Frege1879] p. 52. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege60a (((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))

Proof of Theorem frege60a
StepHypRef Expression
1 frege58acor 36993 . . 3 (((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, (𝜒𝜃), (𝜂𝜁))))
2 ifpimim 36676 . . 3 (if-(𝜑, (𝜒𝜃), (𝜂𝜁)) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁)))
31, 2syl6 34 . 2 (((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜓, 𝜏) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁))))
4 frege12 36930 . 2 ((((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜓, 𝜏) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁)))) → (((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁)))))
53, 4ax-mp 5 1 (((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜒, 𝜂) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, 𝜃, 𝜁))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  if-wif 1005
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 36907  ax-frege2 36908  ax-frege8 36926  ax-frege58a 36992
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-ifp 1006
This theorem is referenced by: (None)
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