Theorem bnj1294

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

	Ref	Expression
Hypotheses	bnj1294.1	$$⊢\phi \to \forall x\in A\psi$$
	bnj1294.2	$$⊢\phi \to x\in A$$
Assertion	bnj1294	$$⊢\phi \to \psi$$

Proof

Step	Hyp	Ref	Expression
1		bnj1294.1	$$⊢\phi \to \forall x\in A\psi$$
2		bnj1294.2	$$⊢\phi \to x\in A$$
3		df-ral	$$⊢\forall x\in A\psi \leftrightarrow \forall x\left(x\in A\to \psi \right)$$
4		sp	$$⊢\forall x\left(x\in A\to \psi \right)\to \left(x\in A\to \psi \right)$$
5	4	impcom	$$⊢\left(x\in A\wedge \forall x\left(x\in A\to \psi \right)\right)\to \psi$$
6	3 5	sylan2b	$$⊢\left(x\in A\wedge \forall x\in A\psi \right)\to \psi$$
7	2 1 6	syl2anc	$$⊢\phi \to \psi$$