Theorem 2rexreu 41185

Description: Double restricted existential uniqueness implies double restricted uniqueness quantification, analogous to 2exeu 2549. (Contributed by Alexander van der Vekens, 25-Jun-2017.)

Assertion

Ref	Expression
2rexreu	|- ( ( E! x e. A E. y e. B ph /\ E! y e. B E. x e. A 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 2rexreu

Step	Hyp	Ref	Expression
1		reurmo 3161	. . . 4 |- ( E! x e. A E. y e. B ph -> E* x e. A E. y e. B ph )
2		reurex 3160	. . . . 5 |- ( E! y e. B ph -> E. y e. B ph )
3	2	rmoimi 41176	. . . 4 |- ( E* x e. A E. y e. B ph -> E* x e. A E! y e. B ph )
4	1, 3	syl 17	. . 3 |- ( E! x e. A E. y e. B ph -> E* x e. A E! y e. B ph )
5		2reurex 41181	. . 3 |- ( E! y e. B E. x e. A ph -> E. x e. A E! y e. B ph )
6	4, 5	anim12ci 591	. 2 |- ( ( E! x e. A E. y e. B ph /\ E! y e. B E. x e. A ph ) -> ( E. x e. A E! y e. B ph /\ E* x e. A E! y e. B ph ) )
7		reu5 3159	. 2 |- ( E! x e. A E! y e. B ph <-> ( E. x e. A E! y e. B ph /\ E* x e. A E! y e. B ph ) )
8	6, 7	sylibr 224	1 |- ( ( E! x e. A E. y e. B ph /\ E! y e. B E. x e. A ph ) -> E! x e. A E! y e. B ph )

Syntax hints:   -> wi 4   /\ wa 384   E.wrex 2913   E!wreu 2914   E*wrmo 2915

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  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920

This theorem is referenced by:  2reu1  41186  2reu2  41187  2reu3  41188