Theorem raaan 2350

Description: Rearrange restricted quantifiers.

Assertion

Ref	Expression
raaan	|- (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps))

Distinct variable groups:   ph,y   ps,x   x,y,A

Proof of Theorem raaan

Step	Hyp	Ref	Expression
1		pm5.1 674	. . 3 |- ((A.x e. A A.y e. A (ph /\ ps) /\ (A.x e. A ph /\ A.y e. A ps)) -> (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps)))
2		rzal 2345	. . 3 |- (A = (/) -> A.x e. A A.y e. A (ph /\ ps))
3		rzal 2345	. . . 4 |- (A = (/) -> A.x e. A ph)
4		rzal 2345	. . . 4 |- (A = (/) -> A.y e. A ps)
5	3, 4	jca 288	. . 3 |- (A = (/) -> (A.x e. A ph /\ A.y e. A ps))
6	1, 2, 5	sylanc 471	. 2 |- (A = (/) -> (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps)))
7		r19.28zv 2340	. . . 4 |- (A =/= (/) -> (A.y e. A (ph /\ ps) <-> (ph /\ A.y e. A ps)))
8	7	ralbidv 1655	. . 3 |- (A =/= (/) -> (A.x e. A A.y e. A (ph /\ ps) <-> A.x e. A (ph /\ A.y e. A ps)))
9		r19.27zv 2343	. . 3 |- (A =/= (/) -> (A.x e. A (ph /\ A.y e. A ps) <-> (A.x e. A ph /\ A.y e. A ps)))
10	8, 9	bitrd 526	. 2 |- (A =/= (/) -> (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps)))
11	6, 10	pm2.61ine 1626	1 |- (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps))

Syntax hints:   <-> wb 146   /\ wa 223   = wceq 953   =/= wne 1577  A.wral 1637  (/)c0 2270

This theorem is referenced by:  cau3ir 6852  climaddlem3 7052  climmullem8 7063  lmcau 7930  hlimcaui 9027

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-v 1803  df-dif 2039  df-nul 2271

Copyright terms: Public domain