Theorem ralidm 2328

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		pm5.1 673	. . 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 2326	. . 3 |- (A = (/) -> A.x e. A A.x e. A ph)
3		rzal 2326	. . 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 2260	. . 3 |- (-. A = (/) <-> E.x x e. A)
6		biimt 728	. . . 4 |- (E.x x e. A -> (A.x e. A ph <-> (E.x x e. A -> A.x e. A ph)))
7		df-ral 1625	. . . . 5 |- (A.x e. A A.x e. A ph <-> A.x(x e. A -> A.x e. A ph))
8		hbra1 1663	. . . . . 6 |- (A.x e. A ph -> A.xA.x e. A ph)
9	8	19.23 1039	. . . . 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 537	. . 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 950  E.wex 956   = wceq 1099   e. wcel 1105  A.wral 1621  (/)c0 2251

This theorem is referenced by:  dfwe2 2898  cnvpo 3463

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 957  df-sb 1155  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-ral 1625  df-v 1787  df-dif 2020  df-nul 2252

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