Theorem exexw

Description: Existential quantification over a given variable is idempotent. Weak version of bj-exexbiex , requiring fewer axioms. (Contributed by Gino Giotto, 4-Nov-2024)

	Ref	Expression
Hypothesis	exexw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
Assertion	exexw	$$⊢\exists x\phi \leftrightarrow \exists x\exists x\phi$$

Proof

Step	Hyp	Ref	Expression
1		exexw.1	$$⊢x=y\to \left(\phi \leftrightarrow \psi \right)$$
2	1	notbid	$$⊢x=y\to \left(\neg \phi \leftrightarrow \neg \psi \right)$$
3	2	hba1w	$$⊢\forall x\neg \phi \to \forall x\forall x\neg \phi$$
4	2	spw	$$⊢\forall x\neg \phi \to \neg \phi$$
5	4	alimi	$$⊢\forall x\forall x\neg \phi \to \forall x\neg \phi$$
6	3 5	impbii	$$⊢\forall x\neg \phi \leftrightarrow \forall x\forall x\neg \phi$$
7	6	notbii	$$⊢\neg \forall x\neg \phi \leftrightarrow \neg \forall x\forall x\neg \phi$$
8		df-ex	$$⊢\exists x\phi \leftrightarrow \neg \forall x\neg \phi$$
9		2exnaln	$$⊢\exists x\exists x\phi \leftrightarrow \neg \forall x\forall x\neg \phi$$
10	7 8 9	3bitr4i	$$⊢\exists x\phi \leftrightarrow \exists x\exists x\phi$$