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Theorem ralidm 3321
 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 nfra1 2355 . . 3 𝑥𝑥𝐴𝑥𝐴 𝜑
2 anidm 376 . . . 4 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
3 rsp2 2371 . . . 4 (∀𝑥𝐴𝑥𝐴 𝜑 → ((𝑥𝐴𝑥𝐴) → 𝜑))
42, 3syl5bir 142 . . 3 (∀𝑥𝐴𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
51, 4ralrimi 2390 . 2 (∀𝑥𝐴𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜑)
6 ax-1 5 . . . 4 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
7 nfra1 2355 . . . . 5 𝑥𝑥𝐴 𝜑
8719.23 1568 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
96, 8sylibr 137 . . 3 (∀𝑥𝐴 𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
10 df-ral 2311 . . 3 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
119, 10sylibr 137 . 2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴𝑥𝐴 𝜑)
125, 11impbii 117 1 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1241  ∃wex 1381   ∈ wcel 1393  ∀wral 2306 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-ial 1427  ax-i5r 1428 This theorem depends on definitions:  df-bi 110  df-nf 1350  df-ral 2311 This theorem is referenced by:  issref  4707  cnvpom  4860
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