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Axiom ax-frege52a 40729
 Description: The case when the content of 𝜑 is identical with the content of 𝜓 and in which a proposition controlled by an element for which we substitute the content of 𝜑 is affirmed (in this specific case the identity logical function) and the same proposition, this time where we substituted the content of 𝜓, is denied does not take place. Part of Axiom 52 of [Frege1879] p. 50. (Contributed by RP, 24-Dec-2019.) (New usage is discouraged.)
Assertion
Ref Expression
ax-frege52a ((𝜑𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))

Detailed syntax breakdown of Axiom ax-frege52a
StepHypRef Expression
1 wph . . 3 wff 𝜑
2 wps . . 3 wff 𝜓
31, 2wb 209 . 2 wff (𝜑𝜓)
4 wth . . . 4 wff 𝜃
5 wch . . . 4 wff 𝜒
61, 4, 5wif 1058 . . 3 wff if-(𝜑, 𝜃, 𝜒)
72, 4, 5wif 1058 . . 3 wff if-(𝜓, 𝜃, 𝜒)
86, 7wi 4 . 2 wff (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒))
93, 8wi 4 1 wff ((𝜑𝜓) → (if-(𝜑, 𝜃, 𝜒) → if-(𝜓, 𝜃, 𝜒)))
 Colors of variables: wff setvar class This axiom is referenced by:  frege52aid  40730  frege53a  40732  frege57a  40745
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