Theorem r19.41vv 3091

Description: Version of r19.41v 3089 with two quantifiers. (Contributed by Thierry Arnoux, 25-Jan-2017.)

Assertion

Ref	Expression
r19.41vv	|- ( E. x e. A E. y e. B ( ph /\ ps ) <-> ( E. x e. A E. y e. B ph /\ ps ) )

Distinct variable groups:   ps, x   ps, y

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

Proof of Theorem r19.41vv

Step	Hyp	Ref	Expression
1		r19.41v 3089	. . 3 |- ( E. y e. B ( ph /\ ps ) <-> ( E. y e. B ph /\ ps ) )
2	1	rexbii 3041	. 2 |- ( E. x e. A E. y e. B ( ph /\ ps ) <-> E. x e. A ( E. y e. B ph /\ ps ) )
3		r19.41v 3089	. 2 |- ( E. x e. A ( E. y e. B ph /\ ps ) <-> ( E. x e. A E. y e. B ph /\ ps ) )
4	2, 3	bitri 264	1 |- ( E. x e. A E. y e. B ( ph /\ ps ) <-> ( E. x e. A E. y e. B ph /\ ps ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wrex 2913

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-rex 2918

This theorem is referenced by:  genpass  9831  dfcgra2  25721  axeuclid  25843  wspthsnwspthsnon  26811  dya2iocnrect  30343  itg2addnclem3  33463