Theorem frege122d

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

	Ref	Expression
Hypotheses	frege122d.a	$$⊢\phi \to A=F\left(X\right)$$
	frege122d.b	$$⊢\phi \to B=F\left(X\right)$$
Assertion	frege122d	$$⊢\phi \to \left(A\mathrm{t+}\left(F\right)B\vee A=B\right)$$

Proof

Step	Hyp	Ref	Expression
1		frege122d.a	$$⊢\phi \to A=F\left(X\right)$$
2		frege122d.b	$$⊢\phi \to B=F\left(X\right)$$
3	1 2	eqtr4d	$$⊢\phi \to A=B$$
4	3	olcd	$$⊢\phi \to \left(A\mathrm{t+}\left(F\right)B\vee A=B\right)$$