# Metamath Proof Explorer

## Theorem r2ex

Description: Double restricted existential quantification. (Contributed by NM, 11-Nov-1995) Reduce dependencies on axioms. (Revised by Wolf Lammen, 10-Jan-2020)

Ref Expression
Assertion r2ex ${⊢}\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}\exists {y}\in {B}\phantom{\rule{.4em}{0ex}}{\phi }↔\exists {x}\phantom{\rule{.4em}{0ex}}\exists {y}\phantom{\rule{.4em}{0ex}}\left(\left({x}\in {A}\wedge {y}\in {B}\right)\wedge {\phi }\right)$

### Proof

Step Hyp Ref Expression
1 r2al ${⊢}\forall {x}\in {A}\phantom{\rule{.4em}{0ex}}\forall {y}\in {B}\phantom{\rule{.4em}{0ex}}¬{\phi }↔\forall {x}\phantom{\rule{.4em}{0ex}}\forall {y}\phantom{\rule{.4em}{0ex}}\left(\left({x}\in {A}\wedge {y}\in {B}\right)\to ¬{\phi }\right)$
2 1 r2exlem ${⊢}\exists {x}\in {A}\phantom{\rule{.4em}{0ex}}\exists {y}\in {B}\phantom{\rule{.4em}{0ex}}{\phi }↔\exists {x}\phantom{\rule{.4em}{0ex}}\exists {y}\phantom{\rule{.4em}{0ex}}\left(\left({x}\in {A}\wedge {y}\in {B}\right)\wedge {\phi }\right)$