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Theorem ralidm 3653
 Description: Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
Assertion
Ref Expression
ralidm (x A x A φx A φ)
Distinct variable group:   x,A
Allowed substitution hint:   φ(x)

Proof of Theorem ralidm
StepHypRef Expression
1 rzal 3651 . . 3 (A = x A x A φ)
2 rzal 3651 . . 3 (A = x A φ)
31, 22thd 231 . 2 (A = → (x A x A φx A φ))
4 neq0 3560 . . 3 A = x x A)
5 biimt 325 . . . 4 (x x A → (x A φ ↔ (x x Ax A φ)))
6 df-ral 2619 . . . . 5 (x A x A φx(x Ax A φ))
7 nfra1 2664 . . . . . 6 xx A φ
8719.23 1801 . . . . 5 (x(x Ax A φ) ↔ (x x Ax A φ))
96, 8bitri 240 . . . 4 (x A x A φ ↔ (x x Ax A φ))
105, 9syl6rbbr 255 . . 3 (x x A → (x A x A φx A φ))
114, 10sylbi 187 . 2 A = → (x A x A φx A φ))
123, 11pm2.61i 156 1 (x A x A φx A φ)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∀wral 2614  ∅c0 3550 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 2478  df-ne 2518  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-dif 3215  df-nul 3551 This theorem is referenced by: (None)
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