Theorem raaan 1775

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 group(s):   ph,y   ps,x   x,y,A

Proof of Theorem raaan

Step	Hyp	Ref	Expression
1		pm5.1 501	. . 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 1773	. . 3 |- (A = (/) -> A.x e. A A.y e. A (ph /\ ps))
3		rzal 1773	. . . 4 |- (A = (/) -> A.x e. A ph)
4		rzal 1773	. . . 4 |- (A = (/) -> A.y e. A ps)
5	3, 4	jca 236	. . 3 |- (A = (/) -> (A.x e. A ph /\ A.y e. A ps))
6	1, 2, 5	sylanc 361	. 2 |- (A = (/) -> (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps)))
7		r19.3rzv 1767	. . . . . 6 |- (-. A = (/) -> (ph <-> A.y e. A ph))
8	7	anbi1d 469	. . . . 5 |- (-. A = (/) -> ((ph /\ A.y e. A ps) <-> (A.y e. A ph /\ A.y e. A ps)))
9		r19.26 1289	. . . . 5 |- (A.y e. A (ph /\ ps) <-> (A.y e. A ph /\ A.y e. A ps))
10	8, 9	syl6rbbr 417	. . . 4 |- (-. A = (/) -> (A.y e. A (ph /\ ps) <-> (ph /\ A.y e. A ps)))
11	10	biraldv 1219	. . 3 |- (-. A = (/) -> (A.x e. A A.y e. A (ph /\ ps) <-> A.x e. A (ph /\ A.y e. A ps)))
12		r19.3rzv 1767	. . . . 5 |- (-. A = (/) -> (A.y e. A ps <-> A.x e. A A.y e. A ps))
13	12	anbi2d 468	. . . 4 |- (-. A = (/) -> ((A.x e. A ph /\ A.y e. A ps) <-> (A.x e. A ph /\ A.x e. A A.y e. A ps)))
14		r19.26 1289	. . . 4 |- (A.x e. A (ph /\ A.y e. A ps) <-> (A.x e. A ph /\ A.x e. A A.y e. A ps))
15	13, 14	syl6rbbr 417	. . 3 |- (-. A = (/) -> (A.x e. A (ph /\ A.y e. A ps) <-> (A.x e. A ph /\ A.y e. A ps)))
16	11, 15	bitrd 406	. 2 |- (-. A = (/) -> (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps)))
17	6, 16	pm2.61i 110	1 |- (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps))

Syntax hints:  -. wn 1   <-> wb 127   /\ wa 196   = wceq 1091  A.wral 1201  (/)c0 1707

This theorem is referenced by:  hlimcaui 5141

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-nul 1708

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