Theorem eean 2181

Description: Rearrange existential quantifiers. (Contributed by NM, 27-Oct-2010.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
eean.1	|- F/ y ph
eean.2	|- F/ x ps

Assertion

Ref	Expression
eean	|- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )

Proof of Theorem eean

Step	Hyp	Ref	Expression
1		eean.1	. . . 4 |- F/ y ph
2	1	19.42 2105	. . 3 |- ( E. y ( ph /\ ps ) <-> ( ph /\ E. y ps ) )
3	2	exbii 1774	. 2 |- ( E. x E. y ( ph /\ ps ) <-> E. x ( ph /\ E. y ps ) )
4		eean.2	. . . 4 |- F/ x ps
5	4	nfex 2154	. . 3 |- F/ x E. y ps
6	5	19.41 2103	. 2 |- ( E. x ( ph /\ E. y ps ) <-> ( E. x ph /\ E. y ps ) )
7	3, 6	bitri 264	1 |- ( E. x E. y ( ph /\ ps ) <-> ( E. x ph /\ E. y ps ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   F/wnf 1708

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-10 2019  ax-11 2034  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710

This theorem is referenced by:  eeanv  2182  reean  3106