Theorem soeq12d

Description: Equality deduction for total orderings. (Contributed by Stefan O'Rear, 19-Jan-2015)

	Ref	Expression
Hypotheses	weeq12d.l	⊢ ( 𝜑 → 𝑅 = 𝑆 )
	weeq12d.r	⊢ ( 𝜑 → 𝐴 = 𝐵 )
Assertion	soeq12d	⊢ ( 𝜑 → ( 𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		weeq12d.l	⊢ ( 𝜑 → 𝑅 = 𝑆 )
2		weeq12d.r	⊢ ( 𝜑 → 𝐴 = 𝐵 )
3		soeq1	⊢ ( 𝑅 = 𝑆 → ( 𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴 ) )
4	1 3	syl	⊢ ( 𝜑 → ( 𝑅 Or 𝐴 ↔ 𝑆 Or 𝐴 ) )
5		soeq2	⊢ ( 𝐴 = 𝐵 → ( 𝑆 Or 𝐴 ↔ 𝑆 Or 𝐵 ) )
6	2 5	syl	⊢ ( 𝜑 → ( 𝑆 Or 𝐴 ↔ 𝑆 Or 𝐵 ) )
7	4 6	bitrd	⊢ ( 𝜑 → ( 𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵 ) )