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Theorem ralidm 4047
Description: Idempotent law for restricted quantifier. (Contributed by NM, 28-Mar-1997.)
Assertion
Ref Expression
ralidm (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ralidm
StepHypRef Expression
1 rzal 4045 . . 3 (𝐴 = ∅ → ∀𝑥𝐴𝑥𝐴 𝜑)
2 rzal 4045 . . 3 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
31, 22thd 255 . 2 (𝐴 = ∅ → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
4 neq0 3906 . . 3 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
5 biimt 350 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑)))
6 df-ral 2912 . . . . 5 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
7 nfra1 2936 . . . . . 6 𝑥𝑥𝐴 𝜑
8719.23 2078 . . . . 5 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
96, 8bitri 264 . . . 4 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
105, 9syl6rbbr 279 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
114, 10sylbi 207 . 2 𝐴 = ∅ → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
123, 11pm2.61i 176 1 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1478   = wceq 1480  wex 1701  wcel 1987  wral 2907  c0 3891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-v 3188  df-dif 3558  df-nul 3892
This theorem is referenced by:  issref  5468  cnvpo  5632  dfwe2  6928
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