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Theorem frege122d 38893
 Description: If 𝐹 is a function, 𝐴 is the successor of 𝑋, and 𝐵 is the successor of 𝑋, then 𝐴 and 𝐵 are the same (or 𝐵 follows 𝐴 in the transitive closure of 𝐹). Similar to Proposition 122 of [Frege1879] p. 79. Compare with frege122 39119. (Contributed by RP, 15-Jul-2020.)
Hypotheses
Ref Expression
frege122d.a (𝜑𝐴 = (𝐹𝑋))
frege122d.b (𝜑𝐵 = (𝐹𝑋))
Assertion
Ref Expression
frege122d (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))

Proof of Theorem frege122d
StepHypRef Expression
1 frege122d.a . . 3 (𝜑𝐴 = (𝐹𝑋))
2 frege122d.b . . 3 (𝜑𝐵 = (𝐹𝑋))
31, 2eqtr4d 2864 . 2 (𝜑𝐴 = 𝐵)
43olcd 907 1 (𝜑 → (𝐴(t+‘𝐹)𝐵𝐴 = 𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∨ wo 880   = wceq 1658   class class class wbr 4873  ‘cfv 6123  t+ctcl 14103 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-ext 2803 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-ex 1881  df-cleq 2818 This theorem is referenced by: (None)
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