Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frege60a Structured version   Visualization version   GIF version

Theorem frege60a 38666
Description: Swap antecedents of ax-frege58a 38663. 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 38664 . . 3 (((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜓, 𝜏) → if-(𝜑, (𝜒𝜃), (𝜂𝜁))))
2 ifpimim 38348 . . 3 (if-(𝜑, (𝜒𝜃), (𝜂𝜁)) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁)))
31, 2syl6 35 . 2 (((𝜓 → (𝜒𝜃)) ∧ (𝜏 → (𝜂𝜁))) → (if-(𝜑, 𝜓, 𝜏) → (if-(𝜑, 𝜒, 𝜂) → if-(𝜑, 𝜃, 𝜁))))
4 frege12 38601 . 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 383  if-wif 1050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-frege1 38578  ax-frege2 38579  ax-frege8 38597  ax-frege58a 38663
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-ifp 1051
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator