Theorem frege122d 38052

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 38279. (Contributed by RP, 15-Jul-2020.)

Hypotheses

Ref	Expression
frege122d.a	|- ( ph -> A = ( F ` X ) )
frege122d.b	|- ( ph -> B = ( F ` X ) )

Assertion

Ref	Expression
frege122d	|- ( ph -> ( A ( t+ ` F ) B \/ A = B ) )

Proof of Theorem frege122d

Step	Hyp	Ref	Expression
1		frege122d.a	. . 3 |- ( ph -> A = ( F ` X ) )
2		frege122d.b	. . 3 |- ( ph -> B = ( F ` X ) )
3	1, 2	eqtr4d 2659	. 2 |- ( ph -> A = B )
4	3	olcd 408	1 |- ( ph -> ( A ( t+ ` F ) B \/ A = B ) )

Syntax hints:   -> wi 4   \/ wo 383   = wceq 1483   class class class wbr 4653   ` cfv 5888   t+ctcl 13724

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-cleq 2615

This theorem is referenced by: (None)