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Theorem ralidm 3349
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 2398 . . 3 𝑥𝑥𝐴𝑥𝐴 𝜑
2 anidm 388 . . . 4 ((𝑥𝐴𝑥𝐴) ↔ 𝑥𝐴)
3 rsp2 2414 . . . 4 (∀𝑥𝐴𝑥𝐴 𝜑 → ((𝑥𝐴𝑥𝐴) → 𝜑))
42, 3syl5bir 151 . . 3 (∀𝑥𝐴𝑥𝐴 𝜑 → (𝑥𝐴𝜑))
51, 4ralrimi 2433 . 2 (∀𝑥𝐴𝑥𝐴 𝜑 → ∀𝑥𝐴 𝜑)
6 ax-1 5 . . . 4 (∀𝑥𝐴 𝜑 → (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
7 nfra1 2398 . . . . 5 𝑥𝑥𝐴 𝜑
8719.23 1609 . . . 4 (∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑) ↔ (∃𝑥 𝑥𝐴 → ∀𝑥𝐴 𝜑))
96, 8sylibr 132 . . 3 (∀𝑥𝐴 𝜑 → ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
10 df-ral 2354 . . 3 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥(𝑥𝐴 → ∀𝑥𝐴 𝜑))
119, 10sylibr 132 . 2 (∀𝑥𝐴 𝜑 → ∀𝑥𝐴𝑥𝐴 𝜑)
125, 11impbii 124 1 (∀𝑥𝐴𝑥𝐴 𝜑 ↔ ∀𝑥𝐴 𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283  wex 1422  wcel 1434  wral 2349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-ial 1468  ax-i5r 1469
This theorem depends on definitions:  df-bi 115  df-nf 1391  df-ral 2354
This theorem is referenced by:  issref  4737  cnvpom  4890
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