Theorem ralanidOLD

Description: Obsolete version of ralanid as of 29-Jun-2023. (Contributed by Peter Mazsa, 30-Dec-2018) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		anclb	$$⊢\left(x\in A\to \phi \right)\leftrightarrow \left(x\in A\to \left(x\in A\wedge \phi \right)\right)$$
2	1	albii	$$⊢\forall x\left(x\in A\to \phi \right)\leftrightarrow \forall x\left(x\in A\to \left(x\in A\wedge \phi \right)\right)$$
3		df-ral	$$⊢\forall x\in A\phi \leftrightarrow \forall x\left(x\in A\to \phi \right)$$
4		df-ral	$$⊢\forall x\in A\left(x\in A\wedge \phi \right)\leftrightarrow \forall x\left(x\in A\to \left(x\in A\wedge \phi \right)\right)$$
5	2 3 4	3bitr4ri	$$⊢\forall x\in A\left(x\in A\wedge \phi \right)\leftrightarrow \forall x\in A\phi$$