Theorem sseqtrid

Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011)

	Ref	Expression
Hypotheses	sseqtrid.1	⊢ 𝐵 ⊆ 𝐴
	sseqtrid.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
Assertion	sseqtrid	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )

Proof

Step	Hyp	Ref	Expression
1		sseqtrid.1	⊢ 𝐵 ⊆ 𝐴
2		sseqtrid.2	⊢ ( 𝜑 → 𝐴 = 𝐶 )
3		sseq2	⊢ ( 𝐴 = 𝐶 → ( 𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐶 ) )
4	3	biimpa	⊢ ( ( 𝐴 = 𝐶 ∧ 𝐵 ⊆ 𝐴 ) → 𝐵 ⊆ 𝐶 )
5	2 1 4	sylancl	⊢ ( 𝜑 → 𝐵 ⊆ 𝐶 )