Theorem rexanidOLD

Description: Obsolete version of rexanid as of 8-Jul-2023. (Contributed by Peter Mazsa, 24-May-2018) (New usage is discouraged.) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		anabs5	$$⊢\left(x\in A\wedge \left(x\in A\wedge \phi \right)\right)\leftrightarrow \left(x\in A\wedge \phi \right)$$
2	1	exbii	$$⊢\exists x\left(x\in A\wedge \left(x\in A\wedge \phi \right)\right)\leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
3		df-rex	$$⊢\exists x\in A\left(x\in A\wedge \phi \right)\leftrightarrow \exists x\left(x\in A\wedge \left(x\in A\wedge \phi \right)\right)$$
4		df-rex	$$⊢\exists x\in A\phi \leftrightarrow \exists x\left(x\in A\wedge \phi \right)$$
5	2 3 4	3bitr4i	$$⊢\exists x\in A\left(x\in A\wedge \phi \right)\leftrightarrow \exists x\in A\phi$$