Theorem frege122d 37071

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 37299. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege122d.a	⊢ (𝜑 → 𝐴 = (𝐹‘𝑋))
frege122d.b	⊢ (𝜑 → 𝐵 = (𝐹‘𝑋))

Assertion

Ref	Expression
frege122d	⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵))

Proof of Theorem frege122d

Step	Hyp	Ref	Expression
1		frege122d.a	. . 3 ⊢ (𝜑 → 𝐴 = (𝐹‘𝑋))
2		frege122d.b	. . 3 ⊢ (𝜑 → 𝐵 = (𝐹‘𝑋))
3	1, 2	eqtr4d 2647	. 2 ⊢ (𝜑 → 𝐴 = 𝐵)
4	3	olcd 407	1 ⊢ (𝜑 → (𝐴(t+‘𝐹)𝐵 ∨ 𝐴 = 𝐵))

Syntax hints:   → wi 4   ∨ wo 382   = wceq 1475   class class class wbr 4583  ‘cfv 5804  t+ctcl 13572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-cleq 2603

This theorem is referenced by: (None)