Theorem raaan 3658

Description: Rearrange restricted quantifiers. (Contributed by NM, 26-Oct-2010.)

Hypotheses

Ref	Expression
raaan.1	|- F/yph
raaan.2	|- F/xps

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:   x,y,A

Allowed substitution hints:   ph(x,y)   ps(x,y)

Proof of Theorem raaan

Step	Hyp	Ref	Expression
1		rzal 3652	. . 3 |- (A = (/) -> A.x e. A A.y e. A (ph /\ ps))
2		rzal 3652	. . 3 |- (A = (/) -> A.x e. A ph)
3		rzal 3652	. . 3 |- (A = (/) -> A.y e. A ps)
4		pm5.1 830	. . 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)))
5	1, 2, 3, 4	syl12anc 1180	. 2 |- (A = (/) -> (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps)))
6		raaan.1	. . . . 5 |- F/yph
7	6	r19.28z 3643	. . . 4 |- (A =/= (/) -> (A.y e. A (ph /\ ps) <-> (ph /\ A.y e. A ps)))
8	7	ralbidv 2635	. . 3 |- (A =/= (/) -> (A.x e. A A.y e. A (ph /\ ps) <-> A.x e. A (ph /\ A.y e. A ps)))
9		nfcv 2490	. . . . 5 |- F/_xA
10		raaan.2	. . . . 5 |- F/xps
11	9, 10	nfral 2668	. . . 4 |- F/xA.y e. A ps
12	11	r19.27z 3649	. . 3 |- (A =/= (/) -> (A.x e. A (ph /\ A.y e. A ps) <-> (A.x e. A ph /\ A.y e. A ps)))
13	8, 12	bitrd 244	. 2 |- (A =/= (/) -> (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps)))
14	5, 13	pm2.61ine 2593	1 |- (A.x e. A A.y e. A (ph /\ ps) <-> (A.x e. A ph /\ A.y e. A ps))

Syntax hints:   <-> wb 176   /\ wa 358   F/wnf 1544   = wceq 1642   =/= wne 2517  A.wral 2615  (/)c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-dif 3216  df-nul 3552

This theorem is referenced by: (None)