Theorem sseq12d

Description: An equality deduction for the subclass relationship. (Contributed by NM, 31-May-1999)

	Ref	Expression
Hypotheses	sseq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
	sseq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
Assertion	sseq12d	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷 ) )

Proof

Step	Hyp	Ref	Expression
1		sseq1d.1	⊢ ( 𝜑 → 𝐴 = 𝐵 )
2		sseq12d.2	⊢ ( 𝜑 → 𝐶 = 𝐷 )
3	1	sseq1d	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶 ) )
4	2	sseq2d	⊢ ( 𝜑 → ( 𝐵 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷 ) )
5	3 4	bitrd	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐷 ) )