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	$$⊢\phi \to A\subseteq B$$
	bnj1241.2	$$⊢\psi \to C=A$$
Assertion	bnj1241	$$⊢\left(\phi \wedge \psi \right)\to C\subseteq B$$

Proof

Step	Hyp	Ref	Expression
1		bnj1241.1	$$⊢\phi \to A\subseteq B$$
2		bnj1241.2	$$⊢\psi \to C=A$$
3	2	eqcomd	$$⊢\psi \to A=C$$
4	3	adantl	$$⊢\left(\phi \wedge \psi \right)\to A=C$$
5	1	adantr	$$⊢\left(\phi \wedge \psi \right)\to A\subseteq B$$
6	4 5	eqsstrrd	$$⊢\left(\phi \wedge \psi \right)\to C\subseteq B$$