Theorem bj-rexvw

Description: A weak version of rexv not using ax-ext (nor df-cleq , df-clel , df-v ), and only core FOL axioms. See also bj-ralvw . (Contributed by BJ, 16-Jun-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-rexvw.1	$$⊢\psi$$
Assertion	bj-rexvw	$$⊢\exists x\in \left\{y|\psi \right\}\phi \leftrightarrow \exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		bj-rexvw.1	$$⊢\psi$$
2		df-rex	$$⊢\exists x\in \left\{y|\psi \right\}\phi \leftrightarrow \exists x\left(x\in \left\{y|\psi \right\}\wedge \phi \right)$$
3	1	vexw	$$⊢x\in \left\{y|\psi \right\}$$
4	3	biantrur	$$⊢\phi \leftrightarrow \left(x\in \left\{y|\psi \right\}\wedge \phi \right)$$
5	4	exbii	$$⊢\exists x\phi \leftrightarrow \exists x\left(x\in \left\{y|\psi \right\}\wedge \phi \right)$$
6	2 5	bitr4i	$$⊢\exists x\in \left\{y|\psi \right\}\phi \leftrightarrow \exists x\phi$$