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Axiom ax-frege52a 37171
 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 funtion ) and the same proposition, this time where we subsituted 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 195 . 2 wff (𝜑𝜓)
4 wth . . . 4 wff 𝜃
5 wch . . . 4 wff 𝜒
61, 4, 5wif 1006 . . 3 wff if-(𝜑, 𝜃, 𝜒)
72, 4, 5wif 1006 . . 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  37172  frege53a  37174  frege57a  37187
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