Theorem eeor 1695

Description: Rearrange existential quantifiers. (Contributed by NM, 8-Aug-1994.)

Hypotheses

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

Assertion

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

Proof of Theorem eeor

Step	Hyp	Ref	Expression
1		eeor.1	. . . 4 |- F/ y ph
2	1	19.45 1683	. . 3 |- ( E. y ( ph \/ ps ) <-> ( ph \/ E. y ps ) )
3	2	exbii 1605	. 2 |- ( E. x E. y ( ph \/ ps ) <-> E. x ( ph \/ E. y ps ) )
4		eeor.2	. . . 4 |- F/ x ps
5	4	nfex 1637	. . 3 |- F/ x E. y ps
6	5	19.44 1682	. 2 |- ( E. x ( ph \/ E. y ps ) <-> ( E. x ph \/ E. y ps ) )
7	3, 6	bitri 184	1 |- ( E. x E. y ( ph \/ ps ) <-> ( E. x ph \/ E. y ps ) )

Syntax hints:   <-> wb 105   \/ wo 708   F/wnf 1460   E.wex 1492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-4 1510  ax-ial 1534

This theorem depends on definitions:  df-bi 117  df-nf 1461

This theorem is referenced by: (None)