Theorem poeq12d

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

	Ref	Expression
Hypotheses	poeq12d.1	⊢ ( 𝜑 → 𝑅 = 𝑆 )
	poeq12d.2	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	poeq12d	⊢ ( 𝜑 → ( 𝑅 Po 𝐴 ↔ 𝑆 Po 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		poeq12d.1	⊢ ( 𝜑 → 𝑅 = 𝑆 )
2		poeq12d.2	⊢ ( 𝜑 → 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	⊢ ( 𝜑 → ( 𝑅 Po 𝐴 ↔ 𝑆 Po 𝐵 ) )