Theorem frege114d 38050

Description: If either R relates A and B or A and B are the same, then either A and B are the same, R relates A and B, R relates B and A. Similar to Proposition 114 of [Frege1879] p. 76. Compare with frege114 38271. (Contributed by RP, 15-Jul-2020.)

Hypothesis

Ref	Expression
frege114d.ab	|- ( ph -> ( A R B \/ A = B ) )

Assertion

Ref	Expression
frege114d	|- ( ph -> ( A R B \/ A = B \/ B R A ) )

Proof of Theorem frege114d

Step	Hyp	Ref	Expression
1		frege114d.ab	. 2 |- ( ph -> ( A R B \/ A = B ) )
2		df-3or 1038	. . . 4 |- ( ( A R B \/ A = B \/ B R A ) <-> ( ( A R B \/ A = B ) \/ B R A ) )
3	2	biimpri 218	. . 3 |- ( ( ( A R B \/ A = B ) \/ B R A ) -> ( A R B \/ A = B \/ B R A ) )
4	3	orcs 409	. 2 |- ( ( A R B \/ A = B ) -> ( A R B \/ A = B \/ B R A ) )
5	1, 4	syl 17	1 |- ( ph -> ( A R B \/ A = B \/ B R A ) )

Syntax hints:   -> wi 4   \/ wo 383   \/ w3o 1036   = wceq 1483   class class class wbr 4653

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 197  df-or 385  df-3or 1038

This theorem is referenced by:  frege111d  38051  frege126d  38054