Theorem e2ebind

Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind is derived from e2ebindVD . (Contributed by Alan Sare, 27-Nov-2014) (Proof modification is discouraged.) (New usage is discouraged.)

	Ref	Expression
Assertion	e2ebind	$$⊢\forall xx=y\to \left(\exists x\exists y\phi \leftrightarrow \exists y\phi \right)$$

Proof

Step	Hyp	Ref	Expression
1		biidd	$$⊢\forall yy=x\to \left(\phi \leftrightarrow \phi \right)$$
2	1	drex1	$$⊢\forall yy=x\to \left(\exists y\phi \leftrightarrow \exists x\phi \right)$$
3	2	drex2	$$⊢\forall yy=x\to \left(\exists y\exists y\phi \leftrightarrow \exists y\exists x\phi \right)$$
4		excom	$$⊢\exists y\exists x\phi \leftrightarrow \exists x\exists y\phi$$
5	3 4	bitrdi	$$⊢\forall yy=x\to \left(\exists y\exists y\phi \leftrightarrow \exists x\exists y\phi \right)$$
6		nfe1	$$⊢Ⅎy\exists y\phi$$
7	6	19.9	$$⊢\exists y\exists y\phi \leftrightarrow \exists y\phi$$
8	5 7	bitr3di	$$⊢\forall yy=x\to \left(\exists x\exists y\phi \leftrightarrow \exists y\phi \right)$$
9	8	aecoms	$$⊢\forall xx=y\to \left(\exists x\exists y\phi \leftrightarrow \exists y\phi \right)$$