Theorem 2reu2rex 41183

Description: Double restricted existential uniqueness, analogous to 2eu2ex 2546. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Assertion

Ref	Expression
2reu2rex	|- ( E! x e. A E! y e. B ph -> E. x e. A E. y e. B ph )

Distinct variable groups:   y, A   x, y   x, B

Allowed substitution hints:   ph( x, y)   A( x)   B( y)

Proof of Theorem 2reu2rex

Step	Hyp	Ref	Expression
1		reurex 3160	. 2 |- ( E! x e. A E! y e. B ph -> E. x e. A E! y e. B ph )
2		reurex 3160	. . 3 |- ( E! y e. B ph -> E. y e. B ph )
3	2	reximi 3011	. 2 |- ( E. x e. A E! y e. B ph -> E. x e. A E. y e. B ph )
4	1, 3	syl 17	1 |- ( E! x e. A E! y e. B ph -> E. x e. A E. y e. B ph )

Syntax hints:   -> wi 4   E.wrex 2913   E!wreu 2914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-eu 2474  df-mo 2475  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920

This theorem is referenced by:  2reu1  41186