Theorem eqab2

Description: Implication of a class abstraction. (Contributed by Peter Mazsa, 16-Apr-2019)

	Ref	Expression
Assertion	eqab2	$$⊢\forall x\left(x\in A\leftrightarrow \phi \right)\to \forall x\in A\phi$$

Proof

Step	Hyp	Ref	Expression
1		biimp	$$⊢\left(x\in A\leftrightarrow \phi \right)\to \left(x\in A\to \phi \right)$$
2	1	alimi	$$⊢\forall x\left(x\in A\leftrightarrow \phi \right)\to \forall x\left(x\in A\to \phi \right)$$
3		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
4	2 3	sylibr	$$⊢\forall x\left(x\in A\leftrightarrow \phi \right)\to \forall x\in A\phi$$