Theorem soeq2

Description: Equality theorem for the strict ordering predicate. (Contributed by NM, 16-Mar-1997)

	Ref	Expression
Assertion	soeq2	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		soss	⊢ ( 𝐴 ⊆ 𝐵 → ( 𝑅 Or 𝐵 → 𝑅 Or 𝐴 ) )
2		soss	⊢ ( 𝐵 ⊆ 𝐴 → ( 𝑅 Or 𝐴 → 𝑅 Or 𝐵 ) )
3	1 2	anim12i	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) → ( ( 𝑅 Or 𝐵 → 𝑅 Or 𝐴 ) ∧ ( 𝑅 Or 𝐴 → 𝑅 Or 𝐵 ) ) )
4		eqss	⊢ ( 𝐴 = 𝐵 ↔ ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴 ) )
5		dfbi2	⊢ ( ( 𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴 ) ↔ ( ( 𝑅 Or 𝐵 → 𝑅 Or 𝐴 ) ∧ ( 𝑅 Or 𝐴 → 𝑅 Or 𝐵 ) ) )
6	3 4 5	3imtr4i	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Or 𝐵 ↔ 𝑅 Or 𝐴 ) )
7	6	bicomd	⊢ ( 𝐴 = 𝐵 → ( 𝑅 Or 𝐴 ↔ 𝑅 Or 𝐵 ) )