Theorem bnj1241

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011) (New usage is discouraged.)

	Ref	Expression
Hypotheses	bnj1241.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
	bnj1241.2	⊢ ( 𝜓 → 𝐶 = 𝐴 )
Assertion	bnj1241	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐶 ⊆ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		bnj1241.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		bnj1241.2	⊢ ( 𝜓 → 𝐶 = 𝐴 )
3	2	eqcomd	⊢ ( 𝜓 → 𝐴 = 𝐶 )
4	3	adantl	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 = 𝐶 )
5	1	adantr	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐴 ⊆ 𝐵 )
6	4 5	eqsstrrd	⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐶 ⊆ 𝐵 )