Theorem ralidm 3654

Description: Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)

Assertion

Ref	Expression
ralidm	|- (A.x e. A A.x e. A ph <-> A.x e. A ph)

Distinct variable group:   x,A

Allowed substitution hint:   ph(x)

Proof of Theorem ralidm

Step	Hyp	Ref	Expression
1		rzal 3652	. . 3 |- (A = (/) -> A.x e. A A.x e. A ph)
2		rzal 3652	. . 3 |- (A = (/) -> A.x e. A ph)
3	1, 2	2thd 231	. 2 |- (A = (/) -> (A.x e. A A.x e. A ph <-> A.x e. A ph))
4		neq0 3561	. . 3 |- (-. A = (/) <-> E.x x e. A)
5		biimt 325	. . . 4 |- (E.x x e. A -> (A.x e. A ph <-> (E.x x e. A -> A.x e. A ph)))
6		df-ral 2620	. . . . 5 |- (A.x e. A A.x e. A ph <-> A.x(x e. A -> A.x e. A ph))
7		nfra1 2665	. . . . . 6 |- F/xA.x e. A ph
8	7	19.23 1801	. . . . 5 |- (A.x(x e. A -> A.x e. A ph) <-> (E.x x e. A -> A.x e. A ph))
9	6, 8	bitri 240	. . . 4 |- (A.x e. A A.x e. A ph <-> (E.x x e. A -> A.x e. A ph))
10	5, 9	syl6rbbr 255	. . 3 |- (E.x x e. A -> (A.x e. A A.x e. A ph <-> A.x e. A ph))
11	4, 10	sylbi 187	. 2 |- (-. A = (/) -> (A.x e. A A.x e. A ph <-> A.x e. A ph))
12	3, 11	pm2.61i 156	1 |- (A.x e. A A.x e. A ph <-> A.x e. A ph)

Syntax hints:  -. wn 3   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642   e. wcel 1710  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)