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Theorem frege67a 37696
Description: Lemma for frege68a 37697. Proposition 67 of [Frege1879] p. 54. (Contributed by RP, 17-Apr-2020.) (Proof modification is discouraged.)
Assertion
Ref Expression
frege67a ((((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒))) → (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))))

Proof of Theorem frege67a
StepHypRef Expression
1 ax-frege58a 37686 . 2 ((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒))
2 frege7 37619 . 2 (((𝜓𝜒) → if-(𝜑, 𝜓, 𝜒)) → ((((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒))) → (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒)))))
31, 2ax-mp 5 1 ((((𝜓𝜒) ↔ 𝜃) → (𝜃 → (𝜓𝜒))) → (((𝜓𝜒) ↔ 𝜃) → (𝜃 → if-(𝜑, 𝜓, 𝜒))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  if-wif 1011
This theorem was proved from axioms:  ax-mp 5  ax-frege1 37601  ax-frege2 37602  ax-frege58a 37686
This theorem is referenced by:  frege68a  37697
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