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Theorem ralidm 4457
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 4455 . . 3 (𝐴 = ∅ → ∀𝑥𝐴𝑥𝐴 𝜑)
2 rzal 4455 . . 3 (𝐴 = ∅ → ∀𝑥𝐴 𝜑)
31, 22thd 266 . 2 (𝐴 = ∅ → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
4 neq0 4312 . . 3 𝐴 = ∅ ↔ ∃𝑥 𝑥𝐴)
5 biimt 362 . . . 4 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑)))
6 df-ral 3147 . . . . 5 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
7 nfra1 3223 . . . . . 6 𝑥𝑥𝐴 𝜑
8719.23 2204 . . . . 5 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
96, 8bitri 276 . . . 4 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
105, 9syl6rbbr 291 . . 3 (∃𝑥 𝑥𝐴 → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
114, 10sylbi 218 . 2 𝐴 = ∅ → (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑))
123, 11pm2.61i 183 1 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wal 1528   = wceq 1530  wex 1773  wcel 2107  wral 3142  c0 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-dif 3942  df-nul 4295
This theorem is referenced by:  cnvpo  6135  dfwe2  7487
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