Theorem raaan 2360

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		pm5.1 676	. . 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 2355	. . 3 |- (A = (/) -> A.x e. A A.y e. A (ph /\ ps))
3		rzal 2355	. . . 4 |- (A = (/) -> A.x e. A ph)
4		rzal 2355	. . . 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 2350	. . . 4 |- (A =/= (/) -> (A.y e. A (ph /\ ps) <-> (ph /\ A.y e. A ps)))
8	7	ralbidv 1663	. . 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 2353	. . 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 528	. 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 1634	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 956   =/= wne 1585  A.wral 1645  (/)c0 2280

This theorem is referenced by:  cau3ir 6915  climaddlem3 7116  climmullem8 7127  lmcau 7996  hlimcaui 9106

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-v 1812  df-dif 2049  df-nul 2281

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