Theorem ralidm 2357

Description: Idempotent law for restricted quantifier.

Assertion

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

Distinct variable group:   x,A

Proof of Theorem ralidm

Step	Hyp	Ref	Expression
1		pm5.1 676	. . 3 |- ((A.x e. A A.x e. A ph /\ A.x e. A ph) -> (A.x e. A A.x e. A ph <-> A.x e. A ph))
2		rzal 2355	. . 3 |- (A = (/) -> A.x e. A A.x e. A ph)
3		rzal 2355	. . 3 |- (A = (/) -> A.x e. A ph)
4	1, 2, 3	sylanc 471	. 2 |- (A = (/) -> (A.x e. A A.x e. A ph <-> A.x e. A ph))
5		n0 2289	. . 3 |- (-. A = (/) <-> E.x x e. A)
6		biimt 731	. . . 4 |- (E.x x e. A -> (A.x e. A ph <-> (E.x x e. A -> A.x e. A ph)))
7		df-ral 1649	. . . . 5 |- (A.x e. A A.x e. A ph <-> A.x(x e. A -> A.x e. A ph))
8		hbra1 1687	. . . . . 6 |- (A.x e. A ph -> A.xA.x e. A ph)
9	8	19.23 1063	. . . . 5 |- (A.x(x e. A -> A.x e. A ph) <-> (E.x x e. A -> A.x e. A ph))
10	7, 9	bitr 173	. . . 4 |- (A.x e. A A.x e. A ph <-> (E.x x e. A -> A.x e. A ph))
11	6, 10	syl6rbbr 539	. . 3 |- (E.x x e. A -> (A.x e. A A.x e. A ph <-> A.x e. A ph))
12	5, 11	sylbi 199	. 2 |- (-. A = (/) -> (A.x e. A A.x e. A ph <-> A.x e. A ph))
13	4, 12	pm2.61i 126	1 |- (A.x e. A A.x e. A ph <-> A.x e. A ph)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146  A.wal 954   = wceq 956   e. wcel 958  E.wex 980  A.wral 1645  (/)c0 2280

This theorem is referenced by:  dfwe2 2935  cnvpo 3522

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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