Theorem poeq12d

Description: Equality deduction for partial orderings. (Contributed by Matthew House, 10-Sep-2025)

	Ref	Expression
Hypotheses	poeq12d.1	|- ( ph -> R = S )
	poeq12d.2	|- ( ph -> A = B )
Assertion	poeq12d	|- ( ph -> ( R Po A <-> S Po B ) )

Proof

Step	Hyp	Ref	Expression
1		poeq12d.1	|- ( ph -> R = S )
2		poeq12d.2	|- ( ph -> A = B )
3		poeq1	|- ( R = S -> ( R Po A <-> S Po A ) )
4		poeq2	|- ( A = B -> ( S Po A <-> S Po B ) )
5	3 4	sylan9bb	|- ( ( R = S /\ A = B ) -> ( R Po A <-> S Po B ) )
6	1 2 5	syl2anc	|- ( ph -> ( R Po A <-> S Po B ) )