Theorem eean 1912

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

Hypotheses

Ref	Expression
eean.1	|- F/yph
eean.2	|- F/xps

Assertion

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

Proof of Theorem eean

Step	Hyp	Ref	Expression
1		eean.1	. . . 4 |- F/yph
2	1	19.42 1880	. . 3 |- (E.y(ph /\ ps) <-> (ph /\ E.yps))
3	2	exbii 1582	. 2 |- (E.xE.y(ph /\ ps) <-> E.x(ph /\ E.yps))
4		eean.2	. . . 4 |- F/xps
5	4	nfex 1843	. . 3 |- F/xE.yps
6	5	19.41 1879	. 2 |- (E.x(ph /\ E.yps) <-> (E.xph /\ E.yps))
7	3, 6	bitri 240	1 |- (E.xE.y(ph /\ ps) <-> (E.xph /\ E.yps))

This proof depends on syntax axioms:   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746

This proof depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545

This theorem is used by:  eeanv  1913  reean  2778