Theorem soeq12d

Description: Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015) (Proof shortened by Matthew House, 10-Sep-2025)

	Ref	Expression
Hypotheses	soeq12d.1	|- ( ph -> R = S )
	soeq12d.2	|- ( ph -> A = B )
Assertion	soeq12d	|- ( ph -> ( R Or A <-> S Or B ) )

Proof

Step	Hyp	Ref	Expression
1		soeq12d.1	|- ( ph -> R = S )
2		soeq12d.2	|- ( ph -> A = B )
3		soeq1	|- ( R = S -> ( R Or A <-> S Or A ) )
4		soeq2	|- ( A = B -> ( S Or A <-> S Or B ) )
5	3 4	sylan9bb	|- ( ( R = S /\ A = B ) -> ( R Or A <-> S Or B ) )
6	1 2 5	syl2anc	|- ( ph -> ( R Or A <-> S Or B ) )