Theorem bj-issetw

Description: The closest one can get to isset without using ax-ext . See also vexw . Note that the only disjoint variable condition is between y and A . From there, one can prove isset using eleq2i (which requires ax-ext and df-cleq ). (Contributed by BJ, 29-Apr-2019) (Proof modification is discouraged.)

	Ref	Expression
Hypothesis	bj-issetw.1	$$⊢\phi$$
Assertion	bj-issetw	$$⊢A\in \left\{x|\phi \right\}\leftrightarrow \exists yy=A$$

Proof

Step	Hyp	Ref	Expression
1		bj-issetw.1	$$⊢\phi$$
2		bj-issetwt	$$⊢\forall x\phi \to \left(A\in \left\{x|\phi \right\}\leftrightarrow \exists yy=A\right)$$
3	2 1	mpg	$$⊢A\in \left\{x|\phi \right\}\leftrightarrow \exists yy=A$$